I have schizophrenia. Here are some of my ideas to break the trend of observation without attention to sufferers' of this, the main form of "madness", own comments.
I used to think that the illness gives you genius, inspired ideas etc, but now I just believe you tend to get wacky, but not particularly deep ideas, like that someone's cast a spell on you with voodoo, or that you are being watched by intelligence agents. There are a million ideas of this kind that schizophrenics get, but the most common single theme is the supernatural. Of course you could say "maybe there are more things in heaven and earth than are dreamt of in your philosophy Horatio..." and suggest that all these delusions are based in fact. I thought this when I first got ill, that I was being spiritually enlightened, I've now softened my stance on that and think that we can all try to get spiritual, but that I'm not (as I thought) the new messiah or anything like that.
If I can think complex or powerful thoughts it's definitely in my capacity as a part-time well person, or using the unharmed functional processes of my brain, because the illness thoughts are just a load of rubbish usually. Also I'd like to say that since genius is nine tenths perspiration, many schizoprenics are out of the running because the illness makes you (seem) lazy. While not genius there can be a beautiful child-like Zen fluidity in art that sufferers of the illness create, sometimes it can be discordant and harsh.
I first got ill when I was doing a Physics/Maths joint honours degree at Bristol Uni (UK), I started to believe the calculus lectures were a secret psychic spy training school, and that people in my student house were talking about me a lot behind my back. I've written 100 pages of a book about my experiences, and have found through this that the most subjective of my strange thoughts are impossible to describe adequately. It's a shame because I want(ed) to communicate my revelations... Still, too bad.
I scored good exam results at school, so you could call me intelligent if you wanted, but I'm less intelligent now because major tranquilisers actually lower your IQ. Quite a lot of my old School and Uni friends thought I had it coming because I thought too deeply, or was just too intelligent, this made me feel quite crap - I am me, I do what I do. Other friends think I am just an eccentric, or a psychological dissenter who believes human nature is different to how mainstream society sees it.
If anyone wants to ask me about my illness I'm happy to talk. (I'm fairly lucid and self-aware most of the time.)