Hodge star computation on a Riemann surface

Question

Let $M$ be a Riemann surface, the Hodge star is defined as $ \alpha \wedge *\beta = \langle \alpha, \overline{\beta} \rangle vol$ where $\langle \cdot , \cdot \rangle$ is the Hermitian product on the complexified cotangent bundle extending the Euclidean one. This definition implies that $*$ is complex linear, and therefore in local coordinates: $*dz= *(dx + idy) = dy + i(*dy) = -i dz$. The problem that I have is that I have read that $*$ maps $(1,0)$-forms to $(0,1)$-forms (in general $*: \Omega^{p,q} \rightarrow \Omega^{n-q,n-p}$) and therefore $*dz$ should be proportional to $d\overline{z}$