We know that for contours $\gamma$ in $\mathbb{C}$, we can define winding numbers as $$ \frac{1}{2\pi i} \oint_\gamma \frac{dz}{z} $$ Are there similar formulation for $\mathbb{C}^2$ or even for all finite-dim complex manifolds? I would imagine that such a closed integral may have possibly non-integer values, but it should be possibly nonzero (in contrast to the real space contour integral $\oint \nabla f \cdot ds = 0$).
EDIT. After some thought, this question may be a little too difficult for a general finite-dim complex manifold. So how about considering such an integral on the 3-sphere in the sense of quaternions, i.e., let $\gamma:[0,1]\to \mathbb{S}^3$ be "well-behaved", then is there a range of values that the integral $$ \Re\left[\frac{1}{2\pi} \oint_\gamma \frac{dz}{iz} \right] $$ takes? Here, a quaternion is written as $z=z_0 +z_1 i+z_2j +z_3k$. Notice by the non-commutative relation of quaternions, $1/iz = 1/z\cdot 1/i$.