Digging up the ages-long discussion about Detective Comics Extended Universe's interpretations of the characters.
So, in the left corner, we have Batman AKA Bruce Wayne. The top-condition middle-aged playboy vigilante who gained a few braincells over his prime counterpart and started killing criminals, fully justifying the police's efforts to start hunting him down. He's seen either killing common thugs in Dawn of Justice or marking them with his logo, basically sentencing them to death by the hands of their co-inmates. The canonical explanation behind his change of heart is that Jason Todd perished at the hands of The Joker, and presumably witnessing The Black Zero accident from Man of Steel.
In the right corner, we have the aforementioned The Joker. The Clown Prince of Crime and The Jester of Genocide, he's a mobster with his own private army of goons who has killed hundreds of people through various means. He has killed Jason Todd the Boy Wonder Robin and turned his psychotherapist into partner-in-crime Harley Quinn. Playing at full gas into What Measure is a Mook? trope, Batman has not killed him yet.
So, what are the theories, canon explanations and headcanons about Batman not killing The Joker?