The Hodge star operator $\ast$ acts on the differential forms of a differential manifold sending $\Omega^{k}$ to $\Omega^{Nk}$. If the manifold is complex, then for $p+q=k$, does $\ast$ map $\Omega^{p,q}$ into some $\Omega^{a,b}$, where $a+b=Nk$.
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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, $$*\Omega^{pq}\subset \Omega^{Nq,Np}$$ where I'm following the notation in the question and writing $\Omega^{pq}$ for the space of $C^\infty$ forms of type $(p,q)$. 

