I have quite a few, truthfully, I'm surprised no one as mentioned them yet.

Angel Heart - Hrmm... the killing only lasts the first few chapters, but its an excellent psych manga, although not listed as such.

MPD Psycho - Now this is the one I am REALLY surprised no one listed. It fits your description perfectly.

SkyHigh - Nothing like that crappy movie of the same name, its really good, and almost meets your description.

Edit- ALSO
Parasyte - Really good. Not really about serial killers, but there are a lot of serial killings in it. Although, it is really a supernatural manga, so you may not like it.

Does the walker choose the path, or the path the walker?

You should read Goth, the protagonist isn't necessarily a serial killer, but he might as well be since he's exactly like one.

Monster is definitely my favourite along this sub-genre. I don't know of anything to recommend. I suppose Doubt is suspenseful enough, but no where near as well done.

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You've probably already read it by now, but if you haven't, I'm here to whole-heartedly third Himitsu, which is excellent and focuses on a police division tasked with solving some of the most heinous crimes in Japan. It's pretty awesome. A handful of the cases have to do with serial murders and one of the main characters was involved with a really grisly serial murder case, so there's some ongoing trauma there. XDD

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neighbor 13 bout a guy with multi personalities and his evil side is a serial killer 😀