If Batman found a broken time travel device, re-made it, traveled through the time stream to a few minutes before his parents were murdered, stopped Chill from killing them by violently punching the thug, but not killing him, went back to the time stream before he vanishes and ended up in a different present time line than the one he'd existed in (with another Bruce there), what would the Dark Knight do next? Btw, this scenario takes place shortly after his first outing as the Bat.
In general? He'd probably just watch over the city or change his name or something, maybe even try to find a way back to his own timeline.
My man most likely goes full stalker mode on his own family - he keeps the Batman identity, still watching over the city and seeing how things are different because he fucked up time like a dummy and now has to see just how bad the damage is, but when he realizes there's another version of himself and his parents are alive still he gets obsessed with just watching them from a distance. He basically stalks them 24/7, following them around from a far distance and just staring like a weirdo.
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As for this AU? He sees how the city has changed, sees the things that are different and what is the same. He sees how he himself has changed. This time, he doesn't stalk his family like a freak because he knows they aren't his. But he does know what is his - his rogues gallery. He hunts them down one by one, seeing what and who they are in this timeline and how their lives are because there is no Batman in this world. Most of them are the same as they were in his time before he stepped in, but some are very different, in both good and bad ways.
Most shockingly: Joker, who is just regular old Jack Napier here. He never fell into the acid, never became Joker, never lost himself. He never became Batman's greatest failure. And the idea of that is something Batman obsesses over - he needs to see what Joker was, who he was. He needs to see what life was like before. Batman has never been able to figure Joker out, the guy has always been a mystery, so now that there was a chance to really solve that case Batman wasn't going to just turn it down.
So, instead of stalking his own family, he stalks Joker (or, Jack Napier, really).
As he watches this poor guy he still tries to figure out how to get back to his own time. By the time he does, he's learned enough about all the rogues - Joker in particular - that he can't really look at them the same. Whenever he looks at them he sees what they could've been if he hadn't stepped in. The mantel of Batman is now stained by the perception of what could've been. He starts to see everything different; he starts to see things the way Joker would. "We're the same, Batsy, you and I. We both know what it's like to suffer, we both have the power to ruin this world, and we could if we really wanted to! But, we don't, because that'd ruin our fun. Our mistakes guide us in this never ending dance. Each turn and each twist. We both know what it's like to ruin. And, in truth, we find comfort in the ruin. We thrive off of it. Isn't that right, Batsy?" (Yeah, rambled coo-coo bullshit, but still somehow poetic I think.) He feels the need to give up the cowl, to start over.
I wanna say this leads to him taking a long look at himself and becoming a new man who is even more bent on saving these people, but we all know character development isn't Bruce Wayne's strong suit. Most likely he gives up and retires the cape and cowl, disbanding the Bat Family and leaving the Justice League, becoming a recluse in Wayne Manor that no one really knows the reason for.
The rogues run rampant on the city until they get bored. Some of them leave, some of them stay and turn to either smaller or larger lives of crime, and others do all they can to get Batman's attention - that's why they became villains, after all, what else do they have?
So, really, no happy endings for anyone! Yippee!
