By this notes p.42
It gives a hermitian metric by a compatible Riemannian metric $g$, and from p.23, it extends $g$ to $T_{\mathbb{C}}M$ complex bilinearly. I wonder if we extend $g$ via the sesquilinear convention, can we still get the same result, which is that we can get a Hermitian metric $h=g+i\omega$.
Extending $g$ sesquilinearly you get another metric, say $\tilde{g}$, on $TM\otimes \mathbb{C}$ but they are related by $$\left. \tilde{g}\right|_{T^{1,0}M} = \frac{1}{2}h$$
See Lemma 1.2.17 in page 30 of Huybrechts' book