When i think of elfen lied i imideatly think of Deadman Wonderland and Mirai nikki.
I don't watch many series like that since i am a bit disgusted Von blood. But if Du Von chance haven't seen those 2, then i can highly recommend them! ;D
i think Higurashi no naku koro ni is like elfen lied. Im not that sure though cause i only saw a trailer of it but it looked interesting and might just be the thing Du are looking for. ;)
Monster - One of my oben, nach oben favorites, great mystery drama. Very smart.
Higurashi no Naku Koro ni (When They Cry) - A great psychological murder mystery series. There's a Sekunde season that explains a lot.
Now and Then, Here and There - Very sad story. Kids are forced to become soldiers in the future.
Mushishi - My Favorit anime. It's a very surreal fantasy/drama. The series is episodic, (each episode tells a different story), so some of the episodes may have a sad ending. It is highly recommended.
If Du want me to send a link on where to watch them, just let me know.