$\tilde{h}$, $g$ and $\omega$ may be $\mathbb{C}$-linearly extended to $TM_\mathbb{C}\times TM_\mathbb{C}\longrightarrow\mathbb{C}$:

and again $\tilde{h}=g-\mathrm{i}\omega$, yet now it makes no sense to say that $g=\text{Re}(\tilde{h})$, $\omega=-\text{Im}(\tilde{h})$ ($g$ and $\omega$ are complex-valued). Moreover, it must be remarked that seeing the complex extension of $\tilde{h}$ as $\tilde{h}:T^{1,0}M\times T^{0,1}M\longrightarrow\mathbb{C}$, then $\tilde{h}=h\circ(\text{id}\times c)$ (it's like adjusting $h$ for it to be $\mathbb{C}$-linear and also tensor).

We note also that

so $\omega$ is a (1,1)-form