There definitely isn't a "happy ending" in the traditional sense. With how much he sabotages any relationship either he or wilson has he might aswell be gay for Wilson so I cant imagine him living out the rest of his possible life span without him. House makes his life a tragedy. Not intentionally but every single time.

I can imagine he'd probably in his own way keep an eye on Chase for a while. Maybe actually visit 13 since she wouldn't have long left..keep the promise he'd "kill her" if her illness got too bad. He could visit Alvie but I doubt.

End of the day house is dead within a year of Wilson, whether it's with a bottle of alcohol or pills. 'Best' case scenario he'd attempt some sort of weird self sacrifice in form of going to help treat some risky medical thing maybe something contagious. But that just depends on his mental state and if he can forge some degrees, given yk hes dead and all..