Regarding the question of whether Jaime Lannister is on a redemption arc, and what that means for his endgame in ASOIAF, I’ve been doing some thinking, and the common thread is basically:
He needs to be redeemed from the twincest. His original sin is having an affair with his sister. Pretty much every villainous thing he’s done can be traced back to his having the bad taste to be involved with Cersei or the bad luck to be Tywin Lannister’s son.
And I’m not blaming Cersei for the times when Jaime behaved like an asshole. She didn’t make him that way. Jaime is the only one responsible for Jaime’s actions. Just that his actions tend to be less shitty when he and Cersei are in different parts of the country. Ergo, he needs to stay the fuck away from Cersei.
He also owes Bran Stark an apology, regardless of how thoroughly he disengages from his sister. And once again, his affair with Cersei was the first and last and primary reason why he chucked Bran out that window. There’s nothing he can do that’ll let Bran walk again, but he can still take responsibility for his actions, and that’s something he needs to do separately from breaking off the twincest relationship.
(But in terms of the character being redeemed; it’s mostly about him getting, and staying, the fuck away from Cersei.)