The following article contains spoilers from "I Am A Gun Finale" from Batman #130, on sale now from DC Comics.
Batman's struggle with his backup personality was the unfortunate result of his commitment to not taking the lives of his enemies. "I Am A Gun Finale" from Batman #130 (by Chip Zdarsky, Leonardo Romero, Jordie Bellaire, and Clayton Cowles) saw Batman wage a war for control of his own mind. Rather than a battle of fists, however, it was one of ideologies. The Batman of Zur-En-Arrh believed that criminals should be executed, but Bruce's natural compassion was resistant to this and won out.
Amid all the chaos, readers received an unexpected piece of information. The true reason why Batman doesn't kill the Joker has been revealed. The Dark Knight believes that there is still some part of the Joker that is worth saving. This belief in humanity and finding it even within Gotham's most depraved criminal proves that Batman is and always will be one of DC's most caring heroes.
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Batman Sees the Humanity in the Joker
While the Batman of Zur-en-Arrh argued for the Joker's death, and even tried to kill him in the real world, Bruce's mind created a construct of the Joker. Or rather, two different versions of him, the one he fights in the present day, and a representation of what Batman believes came before his transformation. This way, Batman could see the full scope of the man he was fighting against, the good and the bad.
In his mind, Batman pictures an ordinary man who was so broken by the world that he became a monster. Far from being happy about it though, Batman recognizes that such an existence must be utter torture. Whoever the Joker was before, he doesn't want the life he's leading now. The litany of crimes his new self perpetrates is just a constant reminder that he is no longer in control, as much as he would like to stop being a monster. So, Batman feels pity and empathy for the man he was, and even though some part of him did agree that the Joker should die, he couldn't bring himself to follow through with it, not if there was a chance that he could save the man he once was.
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Batman Actually Feels Compassion Toward the Joker
As the Joker stated "Batman saves everyone" -- including him. To Batman, all human life is precious, even the ones that are being wasted on villainy. It speaks to the belief in the human spirit Batman has that he can still find something redeemable even in the Joker. Perhaps he suspects that he could do great things if he was somehow rehabilitated. Looking at what he's accomplished as a villain, one could only speculate what being a productive member of society would look like for him.
Or maybe the answer is a lot simpler. Batman sees someone suffering behind the constant smile. A sad, lonely soul who knows that what he's doing is wrong, but has long been gone. That is something Batman can relate to on a very personal level. To have who he was stripped away and replaced with something different, something darker. In the Joker, Batman can still see parallels to his own story and if he can save someone from the misery he feels every day, then it is worth it.