I'm ignorant in regards to what the strict definition of a complex is...
Does a complex require the combination of pulls and presses or just multiple movements?
If the latter is the case, wouldn't a TGU be a complex?
Good point, and it's somewhat semantics and the ways we're used to thinking of things. For example, is "clean + press" (2 movements combined) really that much different than "snatch" (one movement)? It's pretty much the same muscles, same result, and similar effort.
Since get-up is one defined movement/exercise, we wouldn't normally think of it as a complex. But of course, it is a bunch of different movements combined.
Another distinction I've heard (possibly from @Geoff Neupert ?) is "chain" vs. "complex". A kettlebell chain would be clean + press + squat + swing + clean + press. A kettlebell complex would be 3 cleans + 3 presses + 3 squats.
But yes, mainly I think a complex is just combining multiple movements. Doesn't have to require a combination of pulls and presses.