Complex chain rule for complex valued functions

Let $f=f(z)$ and $g=g(w)$ be two complex valued functions which are differentiable in the real sense, $h(z)=g(f(z))$. Prove the complex chain rule. All partial derivatives: $$ \frac{\partial h}{\partial z} = \frac{\partial g}{\partial w}\frac{\partial f}{\partial z} + \frac{\partial g}{\partial \bar w}\frac{\partial \bar f}{\partial z} $$ and $$ \frac{\partial h}{\partial \bar z} = \frac{\partial g}{\partial w}\frac{\partial f}{\partial \bar z} + \frac{\partial g}{\partial \bar w}\frac{\partial\bar f}{\partial \bar z} $$ Are we supposed to arrive at this through Cauchy-Riemann?

This has nothing to do with Cauchy-Riemann's equations. Use the regular chain rule (for functions on $\mathbb{R}^2$) and the definition of the Wirtinger derivatives: $$ \frac{\partial}{\partial z} = \frac12 \left( \frac{\partial}{\partial x} - i\,\frac{\partial}{\partial y} \right) \qquad\text{and}\qquad \frac{\partial}{\partial \bar z} = \frac12 \left( \frac{\partial}{\partial x} + i\,\frac{\partial}{\partial y} \right) $$ It all boils down to a fairly long and tedious algebraic manipulation (See also: Wikipedia)