Gotham City
Should Batman kill the Joker?
In a word, no.
Batman is supposed to be a symbol of justice, but really, he's just a lonely rich guy in a suit. I feel like all of the Batman movies, him fighting crime, are really just Batman trying to convince himself that he's a good guy. The Joker, as twisted as he is, is the perfect outlet for that. As Batman's arch nemesis, Batman loves to fantasize about killing the Joker, ending his regime of terror.
But he never would. Without Joker, Batman's life has no purpose. The Joker is the mastermind behind most of Gotham city's crime. If he were dead, Gotham city would be a lot safer. And what would Batman do then? He'd sit in his batcave, drinking lots of expensive vodka and drowning himself in his sorrows.
And would Joker kill Batman? Also no. Joker wants nothing more than to turn Gotham city into a stage for a brutal video game. If he were to kill Batman, no one in Gotham city would be able to stand up to him. The game loses its fun, its main objective.
What does this mean? This means that Joker and Batman will forever be at an impasse. Fighting eachother, outwitting eachother, but neither will ever truly beat the other. They will fight until nature runs its course.
Now here's a thought:
What if Batman is really just a lonely, delusional man, and in his pain, he imagined a villain to occupy his time? What if the Joker is really just a facet of Batman's personality, and he is both the hero and the villain?