Higurashi is not a horror anime ;__; Well, I know it's so scary but it really is something different than a horror anime. And if you can't understand "the message" you probably will say "what a boring anime." Anyways.
Horror;
Monster - Classic.
Hellsing - Classic and very short. You should watch Hellsing Ultimate too.
Mononoke - Well, it's not that scary but uhh kinda mysteric. Also weird animation style.
Berserk - Classic. Old styled animation.
Another - Currently airing. Maybe you should watch this first.
Elfen Lied - Kinda romantic, kinda dramatic. But it's a "must see" anime. (Vext ;_;)
Ayakashi - Man this is just.. amazing. Horror stories, also includes "Mononoke" chapter.
Gantz - This is just <3 Well I don't know if you like reading manga but you should read its manga. Anime includes first 80 chapters (or so) of manga. Also it's a seinen O/ (and there is a second season)
Shiki - Animation is good. Story is good. Ending is good. Musics are good. Characters are good. I'm sure you will like this one.
Movie suggestions;
Perfect Blue - Satoshi Kon. I can't say anything about this. MUST SEE. MUST MUST SEE.
Paprika - Satoshi Kon + Susumu Hirasawa = Classic. Kinda fantastic, kinda enjoyable and kinda scary. "This is your brain on anime." enough said.