Well, his name is Wilson and he's most certainly not as screwed up as House, but then...
How screwed up is Wilson?
I don't think we get to really see this side of Wilson until the episode with that quote I just used, "House vs God". Almost two complete seasons into the show and only then we get to see Wilson's screwed-up-ness exactly as it is.
Why did the writers take so long to introduce such an important feature of such an important character? Did it occur to them only just then that Wilson should be this way? I don't think so.
One of the reasons I like Wilson better than House is that House is a bit too obvious for my liking, he's too out there, his whole self screams "screwed up" in a way than not even the slowest people could possibly miss. Personally, I like Wilson's subtleness better. We've been getting hints of his issues since the beggining of the show but it was never that clear and open until "House vs God".
I found it pretty damn shocking when I saw the scene where House figures out Wilson had been sleeping with his patient, so didn't expect it from him. Why didn't we all expect it from him? How did we miss such an important trait to his character? Because he has this perfectly nice perfectly well adjusted persona so well developed that the screwed-up-ness is only visible and raw in his most vulnerable momments (for example, having recently been divorced). The only one that could ever see through that carefully developed persona, who would question such niceness is House. We (the viewers) had been just like everyone else on the show, only seeing Wilson like the boy wonder oncologist he appears to be. This is why his screwed-up-ness couldn't be revealed too quickly in the show, because it is supposed to be indeed VERY unobvious.
Why would he sleep with a patient? Because he has a compulsion to be "what whoever he's with needs him to be" (quote from "The Social Contract). Well no wonder he's been divorced three times then, who the hell can keep that up for that long? He pretends - or rather tries really hard - to be this perfect guy who you can count on with anything at any time of the day or of the night or whenever you are in need. Because he needs to save everyone, if he isn't there to save it the world might just fall appart. House interpreted this behaviour for us perfectly in "The Social Contract", it goes back to that one time he wasn't the perfect brother for his brother. His brother ran away and Wilson, "instead of trying to find meaning in semi-random events, he decided never to be that careless again, he developed his people pleasing skills the way an olympic athlete develops his muscles" (more quote). So he works so hard to correct any selfish behaviour of any kind to the point where he wont even bother to buy a water bed when he has wanted one his whole life (until after Amber he always lived either in a hotel room or with someone whom he would let pick the bed just like he did with Amber until she called him on it)
You know how relationships are all about give and take? I'm thinking Wilson does about 90% giving and 10% taking in every relationship he has. His every relationship with every women always seems to start with "She was having a rough time, she needed this or needed that, so I...".
In his relationship with House they usually focus on House's problems, it's him analizing what House does, acting like House's conscience, trying to get House to do the right thing, it's very rarely about Wilson.
He also seems to be very good friends with Cuddy and Cameron, he is there for Cuddy when she just got Rachel, he is there for Cameron in episodes like "Maternity" and "Acceptance" when she was having trouble with her patients.
Do either of them even know that he has a schizophrenic brother? Do either of them know that he was on antidepressants? I don't think so. They weren't there for him through any of that, but hardly their fault if he wont tell them about it. They were there for him when Amber died, that one was hard to miss, but even then did they know what Amber really meant to him? My point is that he is supposedly friends with both of them and he wont bother sharing any personal stuff with them, because every relationship he has he makes a point to make it about the other person and how he can help the other person. How screwed up is that?
When does nice become too nice? I don't think that being too nice will ever be of harm to other people, it can only harm your own self, being nice to the point where you forget any instinct of self preservation and self benefit. We can argue that Wilson's niceness was also harmful to his wives because they married a nice guy persona, like Bonnie said in "House Training" he was so knight in shinning armour. But actually his wives had just as much fault for being seriously retarded, there is no such thing as a knight in shinning armour, they saw Wilson as such because he acted as such and because they were all looking for this perfect guy to come and save them. That is a retarded concept, you are the only one that can save yourself, other people can be of support, but at the end of the day, it's all up to you. There was never any hope for any of Wilson's marriages, no matter wether or not he cheated on them or they cheated on him.
So why House? Why is House the only person with whom Wilson has managed to maintain a long-term connection? Becuase there are very few people in the world who question niceness, and even among those few ones that do I doubt that any of them besides House would go as far as House goes to find out the truth behind it all. Wilson's selfless nature doesn't allow him to share his personal issues with anyone else in order to keep his focus on the problems of whoever he's with.
Wilson didn't bother telling Cuddy or Cameron about being on antidepressants or having a homeless brother, and he didn't bother to tell House either. House himself only found out after stalking him, drugging him, spying on him and much deducing. And I think this is exactly why House might just be the perfect friend for Wilson, or rather the only kind of real friend Wilson can have. No one else would've ever found out about this stuff, because no one would do all the creepy stuff that House does to get to the bottom of things, but apparently it is the only way anyone can ever get to know any personal stuff about Wilson, he certainly doesn't seem willing to volunteer any of this information.
House pretty much sucks at being there for people, specially in delicate momments, he is terribly insensitive and his selfishness and arrogance know no limits, he basically sucks at supporting Wilson in rough times. But at least he knows, and sometimes that is about most of what people really need, to know that there is someone in the world who knows your misery, someone who is aware of what you are going through and to know that they give a crap, even if they aren't very good at showing it and they don't have the right words of support. He hangs on to House because House is the only one who ever sees the real him.
I guess there is no way of measuring screwed-up-ness, so we may never know exactly how screwed up is Wilson, but one thing I know is that the only way a "stupid screwed up friendship" can ever really work is if the members of such a friendship are just as screwed up on their own.
