If we have following identification: $(x,y)\to (z,\overline{z})$
We will have $\frac{\partial}{\partial x}= \frac{\partial}{\partial z}+\frac{\partial}{\partial \overline{z}}$ and $\frac{\partial}{\partial y}= i(\frac{\partial}{\partial z}-\frac{\partial}{\partial \overline{z}})$ Also $dx= \frac{dz+d\overline{z}}{2}, dy= \frac{dz-d\overline{z}}{2i}$
for $f: \mathbb R^2\sim \mathbb C \to \mathbb C$, we have $df= \frac{\partial f}{\partial z} dz+ \frac{\partial f}{\partial \overline{z}} d\overline{z}$
Now Question: I was reading an article, There was one remark: Can someone please explain the following remark. What author intention to make this remark.
Remark: The length $\sqrt{2}$ of $dz$ and $d\overline{z}$, which is imposed by the notation, forces the dual system $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline{z}}$ to have the unnatural length $\frac{1}{\sqrt{2}}$; this is why the chain rule is preferable to duality in their notation.
This remark is Remark 1.1 on page-3, "Complex analysis and CR geometry" book by Giuseppe Zampieri