Consider the following integral over the whole complex plane in $z,\bar z$ $$\int dz \,d\bar z\,\partial_z \,f(z,\bar z)$$ where $f(z,\bar z)$ is some well behaved function.
The two variables $z,\bar z$ are related by complex conjugation $\bar z=(z)^*$, so that we cannot simply 'cancel' the partial derivative and integration in $z$ up to some boundary terms. Still, does the presence of the holomorphic partial derivative $\partial_z$ in the integrand tell us something non-trivial about the integral? Does it simplify somehow?
My own attempt at making sense of this integral is as follows:
For instance, consider transforming the complex variables back to simple euclidean variables via
$$z=x+iy~~,~~\bar z=x-iy$$
Then the integration measure transforms as
$$dz d\bar z=-2idxdy$$
And the holomorphic derivative is
$$\partial_z=\frac{1}{2}(\partial_x-i\partial_y)$$
Plugging this into the integral gives
$$-i\int dx dy(\partial_x-i\partial_y)f(x,y)=-i\int dy[f(x,y)]_{x=-\infty}^{x=\infty}-\int dx[f(x,y)]_{y=-\infty}^{y=\infty}$$
So that indeed the integral kind of simplifies to some boundary terms in some of the components...