> This should generalize easily to the complex/Hermitian case.

This doesn't seem to be true, in that it's actually impossible to have a non-Hermitian matrix C such that x†Cx > 0 over the complex numbers for all x. Whereas over the real numbers, with a matrix R, you can have x'Rx > 0 such that R is asymmetric.

The subtlety here is that x itself can be complex in the complex case, which further constraints C to be Hermitian - see the Wikipedia link I posted above.

In other words, "complex definiteness" is actually a stronger condition than "real definiteness", even for matrices without an imaginary part.