What's the first fundamental form of a regular surface in complex coordinates and how to get it?

Question

Precisely, the first fundamental form of a regular surface is given by $$ds^2=Edx^2+2Fdx\ dy+Gdy^2.$$ What's the form of $ds^2$ in complex coordinates $z=x+iy$.

Answer 1

Summary of comments: after the substitition $dx=(1/2)(dz+d\bar{z})$ and $dy=(1/2)(dz-d\bar{z})$, followed by a copious amount of complex arithmetics, the first fundamental form takes the form $ds^2=\lambda|dz+\mu d\bar z|^2$ where the Beltrami coefficient is $$\mu=\frac{E-G+2Fi}{E+G+2\sqrt{EG-F^2}}$$ and the scaling parameter (I don't think it has a name) is $$\lambda=\frac14(E+G+2\sqrt{EG-F^2})$$

This computation can be found, for example, in lecture notes Geometry of surfaces by T. Rivière.