Although, “Batman:Under the Red Hood” shows how Robin was killed by Joker, But this was some deceiving mystery.
Robin was actually captured by Joker, where he was tortured and brainwashed
After Batman gets rid of Joker, Joker shows off again. But how? Joker is already dead. On invetigation Batman finds that this was new Joker and he was actually the Robin. Robin did not die but he actually thinks he is Joker
When things get out of hand Batman have to get rid of new Joker as well 😉