You think Hannibal Lecter is twisted? How about Norman Bates from Psycho? Think again, they've got nothing on some anime creations.
Anime isn't just big-eyed girls, screaming musclemen and fan-service: it's a medium, filled with many genres for many different audiences. There's dark crime series, supernatural stories, aliens, coming of age stories, medieval fantasies, magic, technologically advanced dystopias... Many even have a combination of several of those elements and more besides. There's something for everyone, and hundreds of launch-points for new fans.
And if you enjoy a good psychological horror but have never actually watched anime before, there are plenty of excellent and entertaining series to watch that'll keep you on the edge of your seat and cater to your tastes. They will satisfy the exact same tastes as good intellectual horrors, leaving the same marks and returning to you late at night. You want scares? You got 'em.
Many of the characters in these series are as complex and disturbing as the greatest horror movie icons and exhibit the same moral grey areas to make you feel conflicted and confused.