Although what I suppose could have happened, would have been maybe a more Spider-Man like thing. Depends on which parent. If it was his mother, then maybe he'd have been egged on by his dad to "avenge your mother". If it was his father, its possible he may still have gone the route to become Batman, but would have his mother to sort of steer him right. With Alfred as his father figure he'd have a sorta normal life... but I could see him either being slightly more of a momma's boy... or more bitter if/when she started dating again and maybe remarried and went on his world tour to get the hell away.
I almost could have seen... had it been just his mother dying... Thomas Wayne becoming a vigilante first then Bruce getting into the act later.
Now one concept I had a few years ago, basically had both parents dying not because of a simple, random mugging, but because Thomas Wayne and Martha were both sort of "socialite vigilantes". And they weren't just mugged in a "gimme the wallet and the jewels." they had a full on brawl and died protecting Bruce. Which later lead Bruce to sort of learn what his folks did and want to continue their mission to keep Gotham cleaner. And most of the gear he had was actually based on things they used and the parents utilized the cave before Bruce ever did.
I think I was thinking of that idea around the time they were rebooting the DCU for the umpteenth time and thought 'well that'd be kinda cool'