i used Norman Bates as an example because he was like a quite unassuming guy, who has an extremely violent side, when the Driver's violent side comes out, yes it is more Punisher-esque, but I was just comparing the 2-sided personality, not what the 2 sides specifically are.
Norman Bates isn't a very good example considering the whole "killed his mother and started killing people in a dress" thing he had going on.
As I said before: everything makes sense when you look at Driver's behavior throughout the movie from the perspective of a child:
1. He's unable to look people in the eye, and answers in one-word responses--like a child would. Also, he seems completely unaware of other people's feelings. This is especially the case where Standard talks/interrogates him, and Driver seems completely oblivious to the fact that Standard is pretty much questioning whether or not Driver slept with Irene
2. Much like some hero from a comic-book, he becomes a literal white knight and serves to protect her. Even when he attempts to ask Irene to run away with him, it's a rather childish way of thinking considering what happened before, and rather than be the bad-ass he tries so desperately to be, he comes off as a sad school-kid awkwardly asking a girl on a date. In a fucked up way, it's kind of adorable how dopeish he comes off as.
3. The most important part to consider is his relationship with Benicio. I do believe that Benicio is meant to be a reflection of Driver's character, and nowhere is this more apparent than the first time Driver visits Irene's apartment. Benicio puts on a Halloween mask, and Driver calls him scary. This image is repeated when Driver himself puts on a mask to kill Nino. Aside from that connection, Benicio is the only character that Driver seems at ease talking to; he's far too unsure of himself to hold up a conversation with another adult, and he's way too love-struck to even dare talk to Irene in a serious manner. Yet, Benicio and Driver seem more like friends than the "father-and-son" relationship one might typically find in crime-fiction.
All of this is integral to understanding why Driver acts as violently as he does. If we're to make a comparison with fellow psycopath Travis Bickle, Bickle's rage is far more focused--Driver's is a literal explosion. When he's pushed to the edge, he lashes out without any real regard for his own safety. In fact, everything that he does is purely reactionary: he doesn't think things through, nor does he seem to have a plan other than protecting Irene and Benicio. He certainly doesn't enjoy his rampage as he's absolutely horrified after the killings at the safe-house and elevator. Driver's violent nature is a part of himself that he can't possibly control. He's a child in a grown man's body.