Do you believe that Batman could exist without the Joker, or vice versa? The Joker has often claimed that Batman needs him so that his mission never ends etc. but my personal opinion is that Batman could function just fine without ol' Uncle J. As for the other way around, I'm not so sure. What do you think? Would he continue to spread madness and death or go into The Dark Knight Returns coma shut-down mode?
I think Joker would probably slowly lose the will to live like he did in The Dark Knight Returns if Batman were to ever bite the dust. He was livid at Sid the Squid in the Batman: The Animated Series episode The Man Who Killed Batman when he thought Sid had accidentally killed Batman and shed an actual tear at the thought of never getting to fight him again. I believe villainy would lose a lot of its romance for him were Batman not around to put a stop to his plans, especially since sometimes Joker does things solely to fuck with Batman (see: Death of the Family).
In The Dark Knight Heath Ledger’s Joker says that Batman completes him, and I think that’s very true. I wish I could remember exactly who said this (I think it may have been Marv Wolfman), but in the documentary Necessary Evil: The Super-Villains of DC Comics a writer says that Joker is sort of the antithesis of Batman and balances out the Batman universe–a hero cannot exist without villains, nor could a villain exist without a hero.
That said, Batman has dozens of enemies so even if Joker were to be wiped from existence he would still serve a purpose as a hero. Bruce places a lot of value on human life, but I do think he’d sleep better at night if Joker was dead.