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Willem Dafoe's version was schizophrenic, rather than someone suffering from dissociative identity disorder (DID). Franco's Harry didn't overdose from anything, though. You're thinking of the comics timeline.

In the comics, Norman originally had DID that functioned similar to its portrayal in The Strange Case of Dr. Jekyll & Mr. Hyde. After Harry overdosed, the writers made Norman more of a psychopath than someone with DID.

With ASM2, the filmmakers could go either way at this point, but I'd lean more towards schizophrenia or balls-out psychopathy.