The "compound verb" here is to have no idea. The syntax is exactly the same with many other verbs - for example, I know what he likes, I cook what he likes, I don't care what he likes.
Note that in some contexts with some verbs, the preposition about can be introduced. Sometimes this simply isn't possible (I cook about what he likes doesn't make much sense). But in other contexts, including about may subtly affect the meaning. Thus...
1: I know what he wants
2: I know about what he wants
...where (1) may simply mean I can identify what he wants. Perhaps we're in an Indian restaurant, and I'm about to tell the waiter that I know my friend wants a Bombay Duck as an appetiser (I may not know anything else about exactly what a Bombay Duck is, but I know that's what he wants).
Note that (2) above is idiomatically unlikely in most contexts, but one that you might encounter occurs in contexts like...
"Shall we go on a boat trip, or sunbathe on the beach? John wants to sunbathe"
3: "Yeah, yeah! We know [all] about what John wants! He just wants to ogle girls in bikinis!"
Even in (3), most people probably wouldn't include [all] about. But at least it doesn't sound "odd".