If you want a really messed up anime, I recommend Happy Sugar Life! Hellsing is a true classic for horror, and I very personally recommend Bokurano. For manga, Drifting Classroom is older but really good, and I also really like Fuan no Tane (or Seeds of Anxiety).

Homunculus by Hideo Yamamoto is deeply and psychologically disturbing. Not for everyone for sure - there's sexual assault - but it's very good. Ichi the Killer is also a fun series, but fucked up--lots of sexual violence and gore here, too, so again, not for everyone.

I second Parasyte; that's easily one of my favorite series.

Anything by Junji Ito. I also like the Kurosagi Corpse Delivery Service, which is about a group of weirdos who search for corpses and ghosts that never got put to rest.

Manhole by Tetsuya Tsutsui is very creepy and has some intense body horror with parasites.

Vinland Saga is neither fantasy nor horror (though it is horrifying historical fiction) is easily one of my favorite series, hands down. If you're into older manga, the Phoenix series by Osamu Tezuka is a solid sci-fi/fantasy read, but jumps between SF and fantasy.

Hands down, my favourite is Houseki no kuni (Land of the Lustrous). On the surface it is about humanoids composed entirely out of gemstones who live in a post-human world. There's lots of emotions and philosophy involved.

Castlevania is pretty good, and so is Corpse Party. Erased is more murder mystery/thriller than horror, but it’s one of the best animes I’ve ever seen. My anime club was watching it every Friday when new episodes were still coming out, and the entire room was standing up and screaming after one particularly bad cliffhanger. If you haven’t seen it, it’s a must-watch