tangent vector on a complex manifold

Question

Let $x_1,x_2, \ldots, x_n$ be local coordinates on a manifolds $M$. One can interpret $\frac{\partial}{\partial x_i}(p)$ as a tangent vetor to a curve with constant $x_j$ (where $j \neq i$). What is the interpretation of $\frac{\partial}{\partial z_i}(p)$ on a complex manifold in local coordinates? I'll be glad for any references.