The whole point of Batman and the Joker is that he would love to kill the Joker! He, as a young boy, watched his parents die right before his eyes! He wants every criminal, every thug that stole his life and the lives of many others to pay! But he's a hero. He won't become the type of monster that he hunts. The moment that he does, he'll never stop. He'll be just another costumed psychopath, except his shtick wouldn't be birds, jokes, hats, cats, or riddles...his would be 'justice'.