Little confusion with complex parameter on surfaces Given a surface $\Sigma$ in $\Bbb{R}^3$ and writing its metric as
$$ds^2= E dx^2+2Fdxdy+Gdy^2$$
we can change its parameter by a complex parameter $z=x+iy$. My confusion is: In the complex parameter $z$, how is written the metric?
I found in some books something like the expression:
$$ds^2=\frac{1}{4}(E-G-2iF)dz^2+\frac{1}{2}(E+G)|dz|^2+\frac{1}{4}(E-G+2iF)d\overline{z}^2$$
For the first and third terms, there is no problem. In fact, I would like to understand the second term.
What's means $|dz|^2$? 
There is no previous mention in the books. So, by the definitions 
$$dx=\frac{1}{2}\big(dz+d\overline{z}\big) \text{  and  }  dy=\frac{1}{2i}\big(dz-d\overline{z}\big)$$
we obtain that
$$4dx^2=dz^2+d\overline{z}^2+dzd\overline{z}+d\overline{z}dz$$
$$-4dy^2=dz^2+d\overline{z}^2-dzd\overline{z}-d\overline{z}dz$$
$$4idxdy=dz^2-d\overline{z}^2-dzd\overline{z}+d\overline{z}dz$$
Following of theses equations clearly the first and third terms.
As there is no true that $dzd\overline{z}=d\overline{z}dz$, then $|dz|^2\neq dzd\overline{z}$. I tried to look this as $|dz|^2=dx^2+dy^2$, because we use that definition when we have isothermal parameters, but unsuccessfully.
I appreciate every help.
 A: It is true that $$|dz|^2 = dzd\bar z = d\bar z dz.$$
[Remember, metrics are symmetric covariant tensors of rank two, and $dz d \bar z$ is really a short way of writing $$\frac 1 2 (dz \otimes d \bar z + d\bar z \otimes dz).$$ This is similar to how $dx dy $ is a short way of  writing $$\frac 1 2 (dx \otimes dy + dy \otimes dx)$$
Your expressions for $4dx^2$, $-4dy^2$ and $4idxdy$ are correct, and if you want to derive them carefully, you would say, for example: \begin{align*} 4dx^2 &:= 4dx \otimes dx \\ &=4\left( \frac 1 2 (dz + d \bar z )\otimes \frac 1 2 (dz + d\bar z )\right)
\\
&=  dz \otimes dz+d \bar z \otimes d \bar z +2.\frac 1 2 (dz \otimes d \bar z + d \bar z \otimes dz)
\\
&:= dz^2 + d \bar z^2+2|dz|^2,\end{align*}
and similarly for $-4dy^2$ and $4idxdy$.]
Anyway, substituting in these expressions for $4dx^2$, $-4dy^2$ and $4idxdy$, we get:
\begin{align*} ds^2 &= Edx^2 + 2Fdxdy + Gdy^2 \\
&= \frac 1 4 E (dz^2 + d \bar z^2 + 2|dz|^2) + \frac{1}{2i} F(dz^2 - d \bar z^2) -\frac 1 4 G(dz^2 + d \bar z^2 - 2 |dz|^2) \\
&= \frac 1 4 (E-G-2iF)dz^2 + \frac 1 2 (E+G)|dz|^2+\frac 1 4 (E-G+2iF)d \bar z^2\end{align*}
