Why Batman is incomplete without Joker
Batman and Joker are the yin and yang of Gotham.
Both of them are crazy and psychotic men who have been shaped by the trauma in their lives.
Joker exists as a constant reminder to Batman as to what he could become if he didn't stick to his principles.
He exists as a check to Batman’s obsession, showing him what would happen if Batman would ever cross the line and turn to the dark side.
And most importantly Joker is the one man who keeps on truly testing Batman. Joker doesn't want to kill Batman. He is like the bully who just wants a reaction out of his victim. Joker wants to drive Batman to the point where he would finally snap and kill Joker. And every time Batman resists this, Batman grows as a character and his worldview and principles evolve and are strengthened by this.
So that is why Batman is incomplete without Joker. Without Joker, Batman would have been just another vigilante with a twisted sense of justice.
Batman is crazy but just the right amount of crazy because of Joker.
This was originally a Quora answer