I am a little muddled and am hoping I can get some clarification about forms in a complex manifold. Since I am only concerned with local issues, consider $M = \mathbb C^n$ as a complex manifold. So I have complex coordinates $z_1,\ldots,z_n$ and corresponding real coordinates $x_1, y_1,\ldots, x_n, y_n$ with $z_j = x_j + i y_j$. Now I have a canonical complex structure $J$ on $M$ given by $J \partial_{x_j} = \partial_{y_j}, J \partial_{y_j} = -\partial_{x_j}$. Then I have an isomorphism of the complex bundles between $TM$ and the holomorphic tangent bundle $T_{(1,0)} M = span_{\mathbb C} \{\partial_{z_j}\}$. This isomorphism is given by $ (a+ib) \partial_{z_j} \mapsto a\partial_{x_j} + b\partial_{y_j}. $ Now let $\omega = \sum_i dx_i \wedge dy_i$ be the standard symplectic form on $M$. What does this correspond to under the above isomorphism. At first glance it is $ \tilde \omega = \frac{i}{2}\sum_j dz_j \wedge d\bar z_j $ where $dz_j = dx_j + i dy_j, d\bar{z_j} = dx_j - i dy_j$. At second glance this can't be correct since this is zero on the holomorphic tangent bundle. On third glance, everything seems to work out if I think of $ d\bar{z_j}(c\partial_{z_j}) = \bar c. $ But this is unsettling to me. Is this the right way to think about it though? Can I get in any trouble by thinking of it this way?
The problem seems to be that the differentials of the real coordinates give real dual vectors, but the covectors $dz_j$, $d\bar z_j$ are inherently complex (their real span is not the real dual to $T_{(1,0)} M$ unless I interpret $d\bar z_j$ as mentioned above).
I believe what is usually done is $\tilde\omega$ is considered to be a form on $TM \otimes \mathbb C$ and is a real symplectic form on $ \mathbb Rspan\{\partial_{x_j}, \partial_{z_j}\} = \mathbb Rspan\{\partial_{z_j} + \partial_{\bar z_j}, i(\partial_{z_j} - \partial_{\bar z_j})\} $ but I'd prefer to not complexify if I don't have to (since I am just concerned with the symplectic geometry and thus the real bundle, but I would like the convenience of the complex notation).