The Kaehler-Ricci flow with complex time

Question

Let $(M,\omega)$ be a Kaehler manifold with Ricci form $\rho$. I define the Kaehler-Ricci flow with complex time by the following equation:

$$\frac{\partial \omega}{\partial z}=\rho$$

with $\frac{\partial \omega}{\partial \bar z}=0$

Is the Kaehler-Ricci flow with complex time defined for complex numbers $z$?