"Are you supposed to be playing with knives, Bats?" note Batman takes up Talia's Katana for the climactic fight with Clayface.
You start off Catwoman's rescue by dropping into a room full of thugs looking to join Two-Face's gang. Said gang numbers at forty-nine (forty-seven unarmed, with Two-Face and a guard in the balcony both armed). If you take out the armed guard first, the second Batman touches down on the main floor, all but six of the mooks run away out of fear.
The opening of the game after Catwoman's intro: not Batman, but Bruce Wayne, is arrested and sent to Arkham City just for trying to lead a campaign to close the place down. Bruce stages an escape with Jack Ryder, then breaks the Penguin's hand while he's still handcuffed before giving a beat down to Penguin's mooks. And that's before Bruce even puts on the Batsuit!
There was one spoiler video showing that you can punch out Penguin as he's cowering from you right after beating the thugs down.
Not only is he capable of beating people up while still handcuffed, but Bruce beats people up so hard, it eventually breaks the handcuffs.
The lead-up to the Demon Trials. Batman needs Ra's al Ghul's blood as a stabilizer for the cure to the poison that is affecting both himself and The Joker. He ends up chasing a member of the League of Assassins throughout Arkham City (a CMOA for the woman herself; before that, she was in the Penguin museum as a trophy because she managed to kill 8 men by herself and Penguin kept her alive because he was impressed by her). After catching up to the woman, Batman places a tracer on her just before she ambushes Bats with the help of more assassins. Batman is saved by Robin — who leaves after giving Batman the Line Launcher to go work on Joker's poisoning scheme — then resumes following the assassin all the way to Arkham City's underground. All of this occurs as Batman becomes sicker and sicker from Joker's blood poisoning.
There's also a subtle awesome moment for Robin. Namely, the fact that he was able to sneak into and out of Arkham City despite the armed TYGER Guards that were positioned to ensure such a thing would not be possible.
Beating the Penguin. He may not be a formidable hand-to-hand combatant, but after what he did to those cops, he definitely deserved it.
The battle with the Joker note or rather, Clayface-masquerading-as-Joker is very fun and satisfying.
After making it to the top of the tower and taking out a few more guards, Batman meets with Hugo Strange, who tells Batman that he "did what you could not," and has saved Gotham. Batman responds by headbutting him and kicking him across the room.
The final battle against Clayface. Scimitar-wielding Batman, anyone?
There's a few different instances of a Battle in the Center of the Mind and they're all awesome, but Ra's al Ghul's is probably the coolest. A hundred illusory ninjas leap at Batman in an attempt to dogpile him... and each has a counter-attack icon above them.
When one guy attacks, you press the button and Batman stops him. When two guys attack, you press the button twice and Batman stops them both at the same time. When three guys attack, you press the button thrice and Batman stops all three at the same time. When ninety-nine guys attack...
Another great moment is how Batman defeats Ra's: Pummeling him, taking his sword and stabbing it into the ground, then slamming his face into the hilt.
If we go by the canon timeline, Batman went into Arkham City and did everything... everything... in a little more than just ten hours.
A moment for Rocksteady: After Arkham Asylum expectations were amazingly high and they not only delivered, but blew the Batmobile out of the water.
Mastering the gliding system, especially with the Grapnel boost upgrade. Once you've gotten the hang of it, you can travel to nearly anywhere, almost never touching the ground.
And according to this, you can actually go a step further and actually leave Arkham City to explore the outside.
Much like Batman: Arkham Asylum, the game is "Throw Batman into a closed circle and watch him take out each member of his Rogues Gallery one by one." If it was a comic miniseries, it would have been pretty cool, but being a video game and giving you the controls to Batman, it gives you a real sense of how big that accomplishment is. Batman isn't awesome just because he's a cunning badass, he's awesome because he refuses to give up. The much larger scale of the game and just how battered he is by the end really drives it home.
Funnily enough, Arkham City does get its own comic miniseries.
More of a Moment of Creepy Awesome, but Hush managed to essentially cut his own face off, harvest Bruce Wayne's facial features from the other inmates, and then operate on himself.
The fight with Freeze. The Awesome Boss for the game. You have to use many tricks in your arsenal because Freeze won't let you try it again.
Freeze: I can evolve my strategies. Can you?
Batman scanning Tyger helicopters to find the master program.
Beating Mad Hatter and his thugs while fighting the influence of one of his mind control devices.
Finally finishing all of the Riddler's challenges and nabbing that condescending bastard. All those frustrating hours are made up for it when you see what happens to him. Before that, you get the oh-so satisfying chance to step on his face and then punch him out.
Most of the Riddler rooms also have some incredible moment in them. In Room 2 (the Shell Game) Riddler cheats. So does Batman.
In Room 3, Riddler brags that he has analyzed your whole arsenal and you have nothing to get past his challenge. So Batman breaks out either Freeze's ice grenades or the zipline instead.
Later, you use the line launcher to slide down a corridor made of spinning blades, do a midair flip and relaunch the zipline at a 90 degree angle!
Room 4 features Batman ziplining over a crusher, grabbing the hostage (who's hanging from the ceiling) and smashing through a window.
Room 5 really does take the cake, however, and is truly impressive for both the Riddler and Batman.
Aaron Cash gets a small one in the final room where Riddler forces the Arkham staff to run continuously unless they want the bombs strapped to their heads to go off. Most of the force is understandably exhausted from all the running, some not even being able to talk past the heavy breathing. But Cash doesn't even sound remotely out of breath, and is still able to intimidate the Riddler during it all.
Cash: I can go all night, Riddler!
Riddler: I'm sure you can, but let's see if you still feel that way tomorrow.
The final fight with Two-Face as Catwoman. It's Nintendo Hard but damn, is it not rewarding. Taking out the goons without them finding the bodies. You can even use a Ledge takedown on Two-Face while he's on the bridge in the middle of the room.
Batman vs. Solomon Grundy. A normal human being managing to take down a 8 ft tall zombie man who's clearly in one of his "Superman would have a bit of trouble here" incarnations.
One for Riddler, who hacks into Strange's feed for no reason other than taunt him. He knows what Strange did at Arkham and he knows what he did to Sharp, but he also knows that the doctor won't be able to trace the broadcast. He's the Insufferable Genius poster boy, and it's glorious.
Strange then utterly one-ups him by revealing that he actually knows Batman's secret identity, and Riddler is left flustered and infuriated because Strange won't tell him.
Riddler: I know you were lying, Strange. There's no way that you could have figured it out! It's some kind of trick. It must be!
Strange: Oh, I use no tricks, no childish puzzles. I simply created a psychological profile of the man most likely to be the Batman, and then matched it against the most logical candidate. I was right, of course.
Riddler: Well, who is he?!
Strange: Ah, but that would spoil the game for you, wouldn't it?
Riddler: You must tell me! I implore you Strange! I-
Strange: Really, Edward? If I could figure it out, it must be child's play for you.
Riddler: But I... I...
Strange: Interesting. Tell me, Edward, how is the Riddler like a blank dictionary?
Strange: You're both at a loss for words.
The Penguin himself gets one. Multiple times, guards, officers and the like tried to get into the Iceberg Lounge to prepare him for transfer to Arkham. They couldn't touch him. When they developed Arkham City, they had to build it around him because he proved completely immovable.
The quest to some political prisoners from being executed by Strange. Batman takes out a veritable 20 or moreElite Mooks in very close quarters who are wielding stun batons and guns, using shields, and are prone to throwing explosive items at him. And Strange is taunting you the entire time. It's an MOA for the player, too, because that fight on anything other than Easy is button-destroyingly difficult, but it feels amazing when you pull it off. The people you rescue are suitably awestruck.
After you complete the main quest, do all the sidquests, save all the political prisoners, solve all the riddles, and do every AR challenge, the loading screens will no longer nag you about what's left to do. Instead they will say, "Arkham site secured and ready for shutdown." It's a little touch, but it give you an amazing sense of accomplishment.
Harley in Arkham City, especially in "Harley Quinn's Revenge", has proved herself to be very competent, managing to capture Batman, of all people.
Harley Quinn's Revenge showcases just how awesome Robin is. He infiltrated Harley's Steel Mill, defeated Harley and several of her armed mooks, beat up a ton of Elite Mooks armed with bats, stun batons, shields, and knives while another was on Titan, and saved Batman.
A small one for the developers. Many games that allow you to play as more than one person just make the alternative characters seem like clones of the original, same fighting style but different moves. Rocksteady gave every character you can play as in this game's DLC's, (namely, Catwoman, Robin, and Nightwing) their own distinct fighting styles , looks, and gadgets, so all of their playing bits feel like fresh, new experiences.
Alfred playing Morality Chain for Batman once Protocol 10 starts, which is revealed to be a massive missle strike to wipe out the criminals in Arkham City. Batman decides to go and save Talia instead of stopping it and Alfred cuts off his access to Wayne Tech satellites and outright refuses to allow him access, firmly stating that Batman cannot let those people die and he must save Gotham.