The Hodge star operator $\ast$ acts on the differential forms of a differential manifold sending $\Omega^{k}$ to $\Omega^{N-k}$. If the manifold is complex, then for $p+q=k$, does $\ast$ map $\Omega^{p,q}$ into some $\Omega^{a,b}$, where $a+b=N-k$. 



Does the Hodge star operator respect complex structure?
As Abtan requested, I'm converting my comments to an answer:


Suppose that $X$ is an $N$ (complex) dimensional complex manifold endowed with a Hermitean metric, or equivalently a Riemannian metric g satisfying $g(JX,JY)=g(X,Y)$, where $J$ is the complex structure.
Let $\*$ denote the $\mathbb{C}$-antilinear extension of the Hodge star operator to complex
valued forms (some people -- including me -- prefer to write this as $\overline{\*}$ as Spiro points out in the comments). Then as one finds on page 82 of Griffiths and Harris,
$$\*\Omega^{pq}\subset \Omega^{N-q,N-p}$$
where I'm following the notation in the question and writing $\Omega^{pq}$ for the
space of $C^\infty$ forms of type $(p,q)$.


