During the confrontation, Jason asks Batman why he didn't execute the Joker after the clown killed him, to which Batman replies, "All I've ever wanted to do is kill him. A day doesn't go by I don't think about subjecting him to every horrendous torture he's dealt out to others, and then end him." The idea of Batman, a man with a genius IQ and Olympic-level physique, fantasizing about torturing someone to death is one of scariest things ever.
Burton's Batman was definitely creepy and all, but he was ultimately an unhinged psycho. Under the Red Hood, though, makes Batman look human, and then twists that humanity into something truly dark. There's even an earlier scene when the Joker talks about how Batman found him after Jason's murder, and put him in a body cast for six months. I've envisioned this scene many times and still can't decide which version is scarier: Batman going all berserk on the Joker, or calmly and methodically breaking every bone in his body.
Although, in my imagination, the Joker doesn't have Jay Leno's chin.
That's why I totally believe animated Batman when he says that if he ever killed, he "will never come back." But Jason doesn't get it. To him, Batman was like a father, but if his death didn't push Batman over the edge, does it mean Bruce never saw him as a son? Of course not. It's just more complicated than that. Then again most normal fathers wouldn't allow their son to leave the house in Speedos instead of actual pants ...
Speaking of family: Under the Red Hood thankfully doesn't make you sit through yet another version of the Waynes getting killed, almost as if Batman was a pop culture icon whose origin is known by pretty much everyone. If that's still not enough for you, then consider this: The movie stars Jensen Ackles from Supernatural as Jason, Neil Patrick Harris as Nightwing, Jason Isaacs as Ra's al Ghul, and John DiMaggio (Bender from Futurama) as a murder-happy Joker.
"Murder is such an ugly word. I prefer extermination. The 'x' makes it sound cool."