Well, the series is over, and a whole lot of stuff happened. Battles, deaths, perhaps a betrayal or two; but now that's over and everyone can build their lives back, right?
Well, not for the person who figures it's too hard to deal with the events of the series. Maybe they were the Well-Intentioned Extremist, or maybe they were controlled by Puppeteer Parasite, or maybe one too many of their loved ones died. Anyhoo, it would take more time than the epilogue has to constructively rebuild with the rest of the cast, so they decide to go Walking the Earth until they feel better about themselves.
Usually with a promise that they and the others "will meet again, someday," assuming the others don't think they're dead. This reunion is rarely ever shown, though if pulled off correctly can result in a very good open ending.
Compare The Atoner.
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Anime and Manga
Tower of God: After successfully making his sister a princess but still feeling dissatisfied, Koon Agero Agnis decided to climb the Tower to search for what he really desired.
Mamimi of FLCL does it at the end by leaving her town and pursuing a career as a photographer.
The overarching plot of X-23's solo series consists of this as Laura goes "walkabout" in the aftermath of her stint on X-Force, in large part triggered by questions over whether she has a soul after an encounter with Hellverine. She leaves Utopia because she feels there are too many voices trying to tell her who she is, and wishes to find her own answers. It's relatively brief, and ends with Laura deciding to attend Avengers Academy after an invitation from Black Widow. During this time, Laura makes substantial progress in at least coming to terms with herself, even if much of her emotional damage remains.
Snufkin does this at the end of the first Moomins book. Granted, he's kinda The Drifter anyway...
He leaves for winter every year. One time, Moomin offered to join him, but was rejected.
Live Action TV
Merrick Baliton does this at the end of the Wild Force season of Power Rangers.
Oz leaves the show with one of these in Buffy the Vampire Slayer to learn to control his werewolf side. He returns for one last episode, seemingly able to control the transformation (not changing to a full moon). He finds his feelings for Willow, who has since moved on, can still trigger it against his will, and leaves for good.
Doctor Franklin did this on Babylon 5. It ended in a near-death hallucination, after getting stabbed during an attempt to intervene in a mugging in Downbelow.
Parodied in the finale of 30 Rock. Liz and Jack have a nice goodbye as he embarks on a yacht trip in the middle of the night to figure out what he wants to do now, but he doesn't get more than a hundred yards before having a brilliant new product idea (see-through dishwashers so you can see what's going on inside) and turning around. "I'm so glad I took that boat trip."
At the end of the seventh season of The Big Bang Theory, Sheldon is feeling overwhelmed by all the changes in his life (the university won't let him change his field of study, Leonard is engaged to Penny and may leave him, the comic book store where he goes burns down) and decides to go riding aimlessly on trains to clear his head.
Sophie Devereaux of Leverage leaves to find her real self partway through season 3, as she realized that none of her aliases were her anymore, despite the depth of backstory that each one had.
Playfully subverted in Lonestar's "No News," when a woman leaves her man to "find herself" and he agonizes over improbable things she must be doing ("Playin' guitar with the band / On the road with Pearl Jam") when the implication is that she's simply left him.
Warhammer 40,000: The Craftworld Eldar live a strictly regimented lifestyle to prevent the hedonism and debauchery that brought down their original civilisation. Some of the younger of their kind who find this too overbearing, or older Eldar lacking a purpose in life, walk the Path of the Outcast: they leave the safety of the Craftworld and wander the galaxy. Some settle on planets and become snipers and hermits, some take to open space in small ships and become Corsairs. The whole point is to experience a little freedom and adventure that they wouldn't be able to experience on the Craftworld. Most get tired eventually and return to their Craftworld, but some don't: some go off and join Harlequin troupes, some stick to the life of piracy, and some embrace hedonism and savagery to such a degree that they seek out the fabled Dark City, Commorragh... And then, of course, some simply don't survive the harsh galaxy and die lost and alone.
Kain from Final Fantasy IV, who has a specific goal in mind — he is going to try to renounce his sins and become a Paladin, as Cecil did in the game.
Leon of Final Fantasy II leaves the party at the end for similar reasons, having been Easily Forgiven for his earlier villainy but considering himself unworthy of forgiveness.
At the end of Phoenix Wright: Ace Attorney, prosecutor Miles Edgeworth, having found out his mentor and a figure he greatly admired were both truly amoral, leaves on a journey to understand true justice. In the second game, Justice For All, he is briefly assumed to be dead, but returns to prosecute the final case, having learned what he sought out to find.
At the end of Suikoden V, Georg leaves to travel the world, setting up his eventual appearance in Suikoden II.
Gilles de Rais, at the end of Jeanne d'Arc, after taking Roger's place in sealing Gilvaroth within himself. If the post-game history sticks to reality, he will not succeed.
Happens in several of the Front Mission games, though best seen in 1st with Royd Clive and 2 with Ash Faruk.
At the end of Lugaru, Turner is Offered the Crown of the rabbit kingdom, but turns it down and chooses to wander the island in search of purpose.
The backstory of The Legend of Zelda: Link's Awakening is that Link left Hyrule to go on a journey of self discovery and training. The game begins with Link returning towards Hyrule at the end of his journey, but along the way he gets shipwrecked and washes ashore on Koholint Island.
Possibly the protagonist of the first Persona game, who we learn in Persona 2: Eternal Punishment had to leave his friends after the end of the first game for some undefined reason. It fits with the development of the other characters from the first game, as while the game involved a lot of "finding oneself," they still had a lot to do between games to actually become the person they realized they were and wanted to be (such as Elly turning from the image-conscious popular girl to the occult nerd she always hid from others). It's strongly implied at the end of Eternal Punishment that the first game's protagonist has finally come back, though we only see the door open on the reunion before he walks in.
Ben of Darths & Droids does this in-between the events of Episode III and Episode IV. However, he makes the mistake of not keeping in contact with everyone and they all think the worse.
An episode of Family Guy ends with Brian going to LA to find himself. The next episode begins with a slew of action movie cliches, none of which even remotely resemble anything that happened in the previous episode. The sequence then ends with Lois exclaming "Brian's gone to LA to find himself!", but even that line didn't appear in the previous episode.
Adventure Time: After defeating the Earl of Lemongrab, Lemonhope decides that, rather than lead the lemon people, he will Walk the Earth to find himself, even if it takes a thousand years. He returns a thousand years later to an abandoned Ooo.