The bloody and death-seeking past of the Red Hood might have one of the most tragic justifications. When looking down the barrel of a gun, Jason Todd revealed a belief that helped explain why he's not afraid to act on his most dark impulses.
When Jason was taken in by Batman, it was clear that he was a much different Robin than the Boy Wonder that preceded him. The young man was bitter at the world and wanted to lash out, and Bruce Wayne thought he could help Jason quell his rage and channel his feelings in a more positive manner. But Todd's anger issues always seemed to get the better of him, causing him to make poor, shortsighted decisions. After coming back from death, Jason donned the identity of Red Hood and continued a violent vigilante streak with seemingly no regard for the consequences.
As it turns out, Jason had a very peculiar thought process which helped him rationalize his killing spree as the Red Hood. In Batman and Robin #4 by Grant Morrison and Philip Tan, Jason returns to Gotham and launches a new war on crime. He ups the ante with this new campaign, taking on a sidekick to assist with his mission and livestream his brutal kills. The two wantonly tear through Gotham's criminal underground, not only getting the attention of Batman and Robin, but from the quickly shrinking pool of crime bosses. To combat the Red Hood, they call on one of the most deadly assassins in the world, Eduardo Flamingo, who proves his worth when he quickly overpowers Jason within minutes of a fight. As the maniacal killer puts a gun to Red Hood's head, the vigilante dares him to do it, saying that even if he dies, he'll come back.
The line has an air of meta-awareness to it, with Jason seeming quite convinced that even if he dies, it'd just be a matter of time until he resurrects. It's not the craziest thought considering death is practically a revolving door in most comic book worlds. But for Red Hood, a belief that he can't truly die would really help explain his loose canon behavior when he first came back from death.
For years, Jason's murder in "Death of the Family" served as a reminder that there were consequences to vigilantism. But by coming back to life, Jason showed that even if the worst-case scenario happens, it's possible to escape it. Without the fears and worries of mortality looming over his head, it's no doubt that Jason would embrace his bloodlust the moment he feels he conquered death. After all, if he did it once, who's to say he wouldn't do it again? Red Hood may not be "immortal" in the traditional sense, but his brush with death does explain his most violent escapades.