How to interpret $\int\mathrm{d}^2z$?

Question

In chapter 6 of Tong's lecture notes on string theory when calculating the Virasoro-Shapiro/4-point Tachyon amplitude he arrives at the integral \begin{align*} C(a, b) = \int\mathrm{d}^2z\ |z|^{2a-2}|1-z|^{2b-2} \end{align*} I can follow his solution in the appendix where he goes on to calculate the integral in terms of the real and imaginary part $z=x+iy$. However, what I don't understand is that he says \begin{align*} \mathrm{d}^2z = 2\mathrm{d}x\,\mathrm{d}y \end{align*} If I think of the statement in terms of differential forms I would naively expect \begin{align*} \mathrm{d}^2z = \mathrm{d}z\,\mathrm{d}\bar{z} = (\mathrm{d}x + i\mathrm{d}y)(\mathrm{d}x-i\mathrm{d}y) = -2i\mathrm{d}x\,\mathrm{d}y \end{align*} What happened to the factor of $-i$? I suppose in the amplitude an extra factor of $-i$ doesn't really matter, but he doesn't mention any such simplifications in his derivation. He also did the same thing when introducing conformal field theories in chapter 4. Does he implicitly include a factor of $\sqrt{-g}$ in the measure? Then I could argue that \begin{align*} \mathrm{d}z\,\mathrm{d}\bar{z}\sqrt{-g^{(z)}} = \mathrm{d}z\,\mathrm{d}\bar{z}\left(-\begin{vmatrix}0 & 1/2 \\ 1/2 & 0\end{vmatrix}\right)^{1/2} = \mathrm{d}z\,\mathrm{d}\bar{z}\left(\frac{i}{2}\right) = \mathrm{d}x\,\mathrm{d}y \end{align*} But now the factor of $2$ is missing.