Motivated by this question, I began to wonder if there is a global definition of the almost complex structure of a complex manifold. It is (almost) always presented as multiplication by complex $i$ on the tangent space, and then globalized. Using the formulae given earlier $$ \overline{\partial}\omega = \frac{1}{2}(\text{d}\omega + i \text{d}(J\omega)), $$ and $$ \partial \omega = \frac{1}{2}(\text{d}\omega  i \text{d}(J\omega)), $$ it is easy to see that $$ \frac{i}{2}d\omega = d(J\omega). $$ Thus, $J\omega = \frac{i}{2}\omega + \omega'$, where $\omega'$ is some closed form. What this $\omega'$ is, however, I cannot see.

closed as no longer relevant by Deane Yang, Tim Perutz, Johannes Ebert, S. Carnahan♦ Feb 24 '11 at 2:17
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Given a complex manifold, you have a bundle of (1,0)forms within complexified 1forms which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives a decomposition of 1forms tensor C into (1,0) and (0,1)part. Your I is an operator which is equal to $\sqrt 1$ on (1,0)forms and $\sqrt 1$ on (0,1)forms. It is in fact real, hence defines a real endomorphism of TM, squared to Id. 

