The way that Shinji seemed, in his psyche, to be undoing all the damage that was done throughout the series, and the way that everyone openly showed how much they cared about him were especially heartwarming; if only End of Evangelion had ended similarly, minus the confusion...
Both endings are canon, so it's possible that Shinji did find peace with himself. The last scene shows that he's still a little troubled, though.
The entire cast congratulating Shinji for his self-realization. Especially when his mother and father simultaneously congratulate him.
To put things straight, the scene before the congratulations shows Shinji rejecting his own personal isolated fantasy world and the world of Instrumentality, which cracks to pieces as he realizes that he can learn to like himself and live in the real world without depending on piloting Eva for other people's approval. He even SAYS as much! That's why the cast (well, the cast that exists in his mind anyway) is congratulating him afterward. It's the most heartwarming moment in the show.
The entire ending exchange:
Kaji: A person's truth is so simple, that most ignore it to concentrate on what they think are deeper truths.
Gendo: You, for example, are simply unused to what it is to be liked by others.
Misato: You've never learned how to deal with fearing what others feel about you and so you avoid it.
Shinji: But, don't the others hate me?
Asuka: What are you, stupid? Haven't you realized it's all in your imagination, you megadork!
Shinji: But, I hate myself.
Rei: One who truly hates himself cannot love, he cannot place his trust in another.
Shinji: I'm a coward. I'm cowardly, sneaky and weak.
Misato: No, only if you think you are, but if you know yourself, you can take care of yourself.
Shinji: [introspectively] I hate myself. But, maybe, maybe I could love myself. Maybe, my life can have a greater value. That's right!. I am no more or less than myself. I am me! I want to be myself! I want to continue existing in this world! My life is worth living here!
[Shinji's room cracks open and he finds himself on a small blue globe with the cast of the series]
Everyone: [applauding] Congratulations!
Shinji: Thank you all!
Shinji telling Rei that she wrings out washcloths like a mother, and Rei blushing
Similarly, when Shinji and Toji go to Rei's apartment to deliver some class bulletins and Shinji starts cleaning up around the place, causing Rei to blush and thank him when she finds out.
Towards the close of episode 10, Magmadiver, for a brief, fleeting moment, we get to see Shinji, for once in his life, able to relax and just enjoy a soak in a hot spring. For someone who is all too frequently tormented for the viewers' entertainment, seeing him genuinely happy for once is truly heartwarming.
It also featured some great moments between Shinji and Asuka. When Asuka is being pulled back out of the volcano, Shinji asks if she's alright, and smiles when she replies with her usual bluster when before he had been annoyed by it. Later. Asuka expects to die as the cord holding her Eva has been severed and she starts falling back into the magma...only to have Shinji in Eva-01 grasp her Eva's hand, causing her to give her first genuine smile of the series and affectionately tell him 'don't over-exert yourself, idiot'. For all the bickering and conflict that arises between these two, they still care about each other a lot.
At the beginning of Episode 7 we see Shinji and Misato squabbling over her slobbish ways and he later complains about her. Then when Misato has to go inside the prototype robot to stop it from rampaging he begs her to come back safely and is extremely worried for her. At the end of the episode, we see Shinji complaining about Misato being a slob again, but this time Kensuke and Toji tell him how stupid he is, because the fact that Misato is comfortable enough to be herself around Shinji shows that even though they were primarily brought together by conflict and unpleasant circumstances, they're family now. The look on Shinji's face as he realizes this is what sells it.
After Asuka's infamous Mind Rape, Asuka is staying at Hikari's house and is not recovering very well, so Hikari tells her that she shouldn't feel bad about herself, and that she and what she has as an EVA pilot done is good and important.
The ending of Episode 11. Though they clashed throughout the episode, and will clash even worse in the future, for one night, on a hill under the stars, Shinji, Asuka, and Rei were actually at peace with one another.
Similarly, the meal they have with Misato at the end of Episode 12. Especially notable because Asuka specifically brought them to eat noodles because she knew Misato couldn't afford a steak dinner, and because Rei said she didn't like meat and Asuka wanted her to be there. It's one of her nicest moments.
In the same episode, Gendo actually congratulates Shinji on a job well done.
Neon Genesis Evangelion Angelic Days Chapter 2-4 has its take on the battle with the Third Angel. Shinji demands to be sent out to battle in place of Kaworu, and ends up in trouble almost right away, forcing Rei to put herself in danger for him, which in turn makes him have to fight for her. After the battle, Shinji sees Rei holding her bandaged hand over her eye. Shinji feels bad for impulsively rushing into battle for her, and feels it's his fault that she got injured. But what does Rei have to say after all this?
Rei:[smiling cheerfully as ever] It bled a lot, but it's no big deal. Shinji:I...I'm... Rei:Shinji. You saved me. Thank you.
In the regular series, Shinji doing all of this would've gotten him into big trouble with Misato. Instead...
Misato:We're not usually good at something the first time we try it, are we? But you did great for your first time, really. You can't blame yourself for what happened to Rei. Shinji:"But I...I was so scared that I couldn't even move! Rei got hurt because she was protecting a coward! And after all that, she still said, "Thank you." Misato:Because you did it. You protected us. All of us.
Heck, just knowing that somewhere out there there is a universe where Shinji, Asuka, and Rei are sane and happy is heartwarming enough.
Rei/Kaworu/Yui's messages in the Instrumentality sequence are pretty heartwarming.
Mainly because it's not Ho Yay for the hell of it that Kaworu's talking about. While Shinji may have been sexually attracted to Kaworu in some odd way, what Kaworu was feeling was pure, unconditional love for someone else and he has no trouble sincerely admitting it. Things get worse sometime after this of course, but Shinji really needed what Kaworu was showing him during that time.
Yui's soul making contact with Shinji near the end of Episode 16. The scene just feels so warm and loving, like a mother should be.
Misato sobbing uncontrollably at the sight of Shinji unharmed, then hugging him tightly in Episode 16 proved somebody genuinely cares about Shinji after all.
Even better since when he wakes up in the hospital, Rei is there next to the bed... and Asuka was hiding just outside the door.
Knowing that although the Gainax Ending looks bleak, there is now a new hope to humanity. Shinji and Asuka might be the only souls left on Earth, but there's still billions more people who can come back if they wish.
Hikari sitting next to Touji's bedside. That girl definitely loves the guy, and after what he's gone through, he deserves her love.
Throughout all of episode 19, Misato. Her desire to have Shinji back, cursing out Ritsuko and her "damn science" for not being really helpful at all...she really loved Shinji.
The moment when Shinji grieves over Kaworu.
In End of Evangelion:
Misato's smile as she sends off Shinji for the last time.
Right before Maya turns into LCL, Ritsuko appears, hugs her and quickly types something on Maya's laptop, which is sitting next to them. Then we get a shot of the screen as the LCL splashes on it, and written on it is "I Need You".
If you can get past the minimalisticcreepiness of the epilogue, there's a subtle but powerful moment in the Bittersweet Ending. After Asuka comes out of the Sea of LCL, Shinji, at the edge of his sanity, strangles her. So rather than to return the favor, Asuka does the impossible: she caresses his face. After this, he stops, realizes what he's doing and releases her, and chooses to stay by her side after regaining his composure (as shown by the poster art associated with the film, which is canon according to Anno). This may double as an Aww, Look! They Really Do Love Each Other moment, but it's hard to tell for some people due to the overall tone of the film.
The most common and sensible conclusion concerning the ending (which we can call official) is that Shinji strangled Asuka to confirm that he had returned to reality. To put it into emphasis, following Third Impact most people had been turned into LCL and the oceans of LCL flooding the Earth is the rest of humanity. When Rei merged with Lilith and started it the souls of humans were gathered and everyone is given the world that they desire. In short: the mother of all LotusEaterMachine plots When Shinji rejected it, Rei/Lilith dies and is no longer able to hold them in her possession. Following that everyone whose soul was held in her hand above the planet returned to Earth. The soul along with LCL which floated away after being turned into LCL had returned to them. Shinji speaks with the human avatar representatives of both the souls of Adam and Lilith (Kaworu and Rei) who want to confirm if Shinji is certain about returning the way reality originally was. As Evangelion Unit 01 regains control and the mass evas links are severed. As Third Impact ends Shinji meets with his mom who careses his cheek and asks him if he will be okay. Shinji assures her that he will and then they both go their separate ways. Shinji back to Earth and Yui's soul (Eva Unit 01) leaves along with the Lance of Longinus goes through space. In the final: One More I need you. Shinji finds himself in the sea of LCL and following some time passed he heads to shore. He wakes up to find himself on shore and sees Asuka who appears to be in quite the weakened state. Shinji mimicking his actions during Third Impact (13:25). Asuka however, not as idle as the one he speaks with during Third Impact and rather than responding in her regular Tsundere manner she does the most wildly unexpected: caresses his cheek just as his mother did not hours ago before he arrived on shore.◊ Shinji then breaks down and cries on her. Asuka in Mood Dissonance simply says "I feel sick(Original)/"How Disgusting"(English dub) in the most quiet voice imaginable before the movie ends.
"I'm only asking you to believe this: If...I mean, no matter what happens to me, I will not be sorry to die, because I had that time with you."
Shinji and Rei's little heart to heart. It really shows how well they get along with each other, with Shinji opening up to her while Rei responding and actually making the first move in inviting him into her apartment for tea. If this wasn't Evangelion, they really would have become great friends.
Later on in volume 9: "Could I...hold your hand again?"
Rei's Heroic Sacrifice to save Shinji is much more played up in the manga, and it's just as heartwarming as it is tear-jerking.
Asuka and Shinji practicing the dance routine to defeat Israfel together]]. After squabbling and fighting with each other ever since they met, they finally reach an understanding. The little smile they share with each other as they perfect the routine is adorable.
Asuka telling Shinji that he did pretty well 'for a klutz', then both of them falling asleep like exhausted puppies.
Shinji saving Asuka from what was her cruel fate at the hands of the MP-Evas in End of Evangelion, and Asuka being grateful and smiling at him for it, showcasing Character Development from before where she would have been pissed off and embarrassed at being saved.
The ending, particularly the epilogue. It may be a Bittersweet Ending, but it's as happy a one as Evangelion can have.