Haibane Renmei is a nice suggestion, but that's to be expected from jedinat.
It has 13 episodes, and as far as I know there is only 1 series/season. For this and more info you could have a look at anidb:
http://anidb.info/perl-bin/animedb.p...=anime&aid=271
Personally the words 'supernatural' and 'psychological' made me thinks of Serial Experiments Lain. Though perhaps it is more technological-supernatural and philisophical-psychological.
(One of those annoying/great animes that makes you think about stuff, rather than just drool with brains in decay-mode until there's an unambigious happy end.) ^_^
Anidb entry:
http://anidb.info/perl-bin/animedb.pl?show=anime&aid=99