What anime(and manga) do you recommend to someone, who likes Death Note ?

Code Geass, definitely. Lelouch and Light are very similar characters, although they're also very different (Lelouch is less sociopathic than Light--he actually cares about other people). It also starts out with the same premise of "guy gains special powers and starts using them to change the world." Plus, it's an amazing show. =)

I'm not sure. I love Death Note, I love Bleach and Naruto but I wouldn't say they were similar to DN. If you don't mind gory stuff, Hellsing's good, but Hellsing Ultimate (they're the newer ones, and far more close to the manga). Just realised I'm possibly not the best to answer this, but I tried ;) Hope it helped

Bleach is cool i reccomend Code Geass theres alot going on....there is romance...[in bleach theres a good chunk of action+romance alot of blood depending on who is fighting so..]

Definitely if you like more mature and deep-plotted anime, Monster is amazing:
link

I would also recommend Bleach, Darker Than BLACK, Gantz, Detective Conan, Spiral, ChaoS;HEAd, Switch, Claymore, Ergo Proxy, Hell Girl, Jigoku Shoujo, Eureka Seven, Descendants of Darkness, Higurashi no Naku Koro ni, Serial Experiment: Lain.

None of those are similar to "Death Note" in every aspect, because Death Note is a very complex and unique concept and amazing anime/manga.
They all have some similarity though, be it battle-of-the-wits, out-doing the police, psychological games, a deep plot, shinigami or some sort of aspect like that.
:)

Well I'm not sure! I love Death Note so I'll just tell you mangas that I know and like but I can't assure you you'll like them all! So there is: Naruto (anime and manga), Zombie Powder (manga), Kyo (manga), Until Death do us part (manga), Bleach (anime and manga), fruits basket (manga), Blue dragon (anime and manga) and I think this is it! Btw, the ones that I only wrote manga is only because I don't know if they made animes of them... and again btw, these mangas aren't all from the same style as Death Note, I just wrote the mangas I like cuz I'm not sure which ones can be ''for someone who loves Death Note''!