Can a possessing fiend target the possesed creature with "self" range spell like abilities?
For example an advanced (4 HD) Quasit is possessing a human peasant. Could the Quasit use it's
Invisibility on the possessed character?
Yes, if the fiend acts as a controller. Eberron Campaign Setting, page 101:
This is assuming the fiend doesn't lose his own spell-like abilities when he possesses someone. As the rules don't say he loses them, he doesn't.
Interesting question. It's obvious that someone possessing someone would lose their Ex abilities, as those are more physical attributes.
But they keep their spellcasting, so I think they would probably thus keep their Sp abilities. In the quasit's case, the invisibility is just a spell-like lumped in with the others, so I would say they definitely get it, and "self" would be their containing body.
But what about Su abilities? I'd tend to rule no, unless it entertained me as the GM.
Sorry for raising this question from the dead, but I think the existing answer may be wrong, depending on what sources you're using. I don't have Eberron Campaign Setting, but the Fiendish Codex I: Hordes of the Abyss is pretty clear on this. Before possessing a creature, the fiend must "shuck its physical form and take on an incorporeal one that enables it to possess other creatures." So, a possessing fiend is physically present elsewhere when it is possessing a creature via its incorporeal form.
Fiendish Codex I: Hordes of the Abyss, Page 22.
So, no spell-like abilities, self-only or otherwise.