This story can be in both setting. In Present Gotham With Bruce Wayne batman. a new masked character has appeared. with a display of very advanced technology the woman is causing all sorts of confusion with a play on the 'robin hood' Idea. stealing from the rich and greedy and dispensing it among the less fortunate. on top of that if the rich or influential try to get what was theirs back, she's popping up to fight them off as if she already knows exactly wear and when it'll happen. Who is this mysterious woman and how did she get her hands on such amazing technology? That's what your character wants to find out.
In Future Gotham. Terry McGinnis Batman. has been batman for a while now. starting his first years of collage he meets a Girl named Caroline and they hit it off. but he also meets a new Villein, Circuit Breaker. A self proclaimed robin hood she carries a lot of powerful tech and even adapts to changes made to the bat suit. You character again is curious about this new masked character, who is she, what is her ultimate goal?
((These plots are open to any character be it Joker, Bruce!Batman, Terry!Batman, or even a revived Mr. Freeze, It's totally up to you Shoot me a PM if your interested :)))