There is a lot of already good recommendations in this thread so ill just add some that i feel deserve some credit.
Not really comprehensive at all but just some shows i had just recently watched for the first time:
Strange Dawn - Thought it was a good adventure/fantasy type anime, it was enjoyable to watch and didn't feel stupid. Despite its appearance its actually quite good. Only thing i didnt like about this show was the abrupt end. Probably not on Netflix.
Serial Experiments Lain - Interesting to watch and to interpret, probably one of the more original anime i've seen as far as storytelling goes and also it can be confusing at first. I believe Netflix has the DVDs but not on their streaming service.
Perfect Blue - Not a series, but its one of the better psychological-thriller anime available. I believe it is on netflix streaming and i cant recommend this film enough.
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