Integration on Complex Manifolds

Question

Let $M$ be a complex manifold. Then there are three different notions of a tangent space on $M$;

1. The real tangent space locally generated (on some coordinate patch) over smooth real functions locally on a coordinate patch by $\partial_{x}, \partial_y$

2. The complexified tangent space locally generated over smooth complex functions by $\partial_z, \partial_{\overline{z}}$

3. The holomorphic tangent space locally generated over holomorphic function by $\partial_z$

Then, dually, there exist three types of cotangent spaces on a complex manifold.

Roughly, in the smooth case, if $M$ has dimension $n$, then we integrate over $M$ by integrating (global) sections of $\Lambda^n T_M^*$.

In the complex case, what is the correct notion of integration?

It seems like we have a few options here.