Neon Genesis Evangelion, NHK ni Youkoso! Recommendations
Both have themes about loneliness and are great psychological dramas.
If that's what your looking for you should definitely watch NGE. That's only one of many reasons though.
The premier anime psychological drama. Welcome to the NHK takes many cues from the Evangelion school of psychological exposition and character deconstruction, and any viewer with a taste for such things would do well to check out both.
Both are about main character's spiritual wounds.
Based on psychological and social themes.
At their core, Welcome to the NHK and Neon Genesis Evangelion are both criticisms of otaku culture and the lonely anti-social existance that otaku live. (Although NHK is more about hikkikomori, it still has quite a lot to say about otaku.)
NHK gives an accurate portrayal of hikkikomori/otaku life; if you are at all anti-social, you'll understand that the characters of NHK are among the most realistic you'll find from television. It certainly helps that the author of the original light novel was a hikkikomori himself.
Evangelion on the other hand, is just huge ball of contempt and anger directed at otaku culture,
The themes of self-worth, escapism and communication are explored in both of these shows.
If you enjoyed the psychological elements of Neon Genesis Evangelion, this is a great next stop.
Both are psychological anime that deal with depression. The main character in both have a strong desire to 'run away' from the problems facing them. Neon Genesis Evangelion is considerably harder to digest than NHK. It is recommended both be watched without reading to much about them, both are good, just go watch them.
Both deal with psychological issues, where NGE is more gritty, Welcome the NHK deals with troubles in a more comedic style but dealing with similar issues. If you enjoyed analyzing characters in NGE, the characters in NHK will have some similar issues for you to discover.
Both anime contain subjects, like depression, loneliness, portrayed by a screwed-up and emotionally broken character cast. NHK takes a rather realistic approach, even if both can become extremly depressing.