Do you agree that something like $z_1^2z_2 dz_1 \wedge d\bar{z}_3 + z_3 dz_1 \wedge dz_2$ is a natural sort of thing to integrate over a complex 2 dimensional submanifold of $C^3$? I think that these forms are "intrinsically interesting", just because we want to do integral calculus.
Closed forms are interesting because their integrals are $0$. Closed forms are always locally exact, but may or may not be globally exact. It seems intrinsically interesting to investigate those forms which are locally but not globally exact.
I guess I am just saying that these are the natural questions you would ask when you start doing integral calculus on complex spaces?
What sort of application are you looking for?