Code Geass and Mob Psycho 100 are great anime similar to Death Note, all with complex protagonists and themes. Psycho Pass is the closest anime to Death Note, focusing on conspiracies and justice, but ...

First up to bat is Code Geass which, in itself, is also about a megalomaniacal genius who uses underhanded methods to try and change the world for the better as an idealistic rival tries to stop him.

Anime like Death Note simply do not come around often. Psychological series are rare in general, let alone cat-and-mouse stories that can rival an all-time great show that transcends manga or anime.