In equation 5.37 of francesco's CFT he writes the Ward Identities for traslation symmetry in the language of holomorphic functions. He goes from
\begin{equation} \frac{\partial}{\partial x^\mu} \langle T^\mu_{\ \ \ \nu}(x) X\rangle =-\sum_i^N\delta(x-x_i)\frac{\partial}{\partial x_i^\nu}\langle X\rangle \tag{5.32a}\end{equation}
to
\begin{align} 2\pi\partial_\bar{z}\langle T_{zz}X\rangle+2\pi\partial_z\langle T_{\bar{z}z}X\rangle=-\sum_i^N\partial_{\bar{z}}\frac{1}{z-\omega_i}\partial_{\omega_i}\langle X\rangle \tag{5.37a} \\ 2\pi\partial_\bar{z}\langle T_{z\bar{z}}X\rangle+2\pi\partial_z\langle T_{\bar{z}\bar{z}}X\rangle=-\sum_i^N\partial_{z}\frac{1}{\bar{z}-\bar{\omega}_i}\partial_{\bar{\omega}_i}\langle X\rangle. \tag{5.37b} \end{align}
I understand how to write the delta function as
\begin{equation} \delta(x-x_i)=\frac{1}{\pi}\partial_z\frac{1}{\bar{z}-\bar{\omega}_i}=\frac{1}{\pi}\partial_\bar{z}\frac{1}{z-\omega_i}\tag{5.33} \end{equation}
but the index manipulation on the left hand side is very obscure to me. When I try to perform the change of variables explicitly I get
\begin{align} \frac{\partial}{\partial x^\mu}T^\mu_{\ \ \ \nu}&=\frac{\partial}{\partial x^1} T^1_{\ \ \ \nu}+\frac{\partial}{\partial x^2} T^2_{\ \ \ \nu} \\ &= \Big\{\frac{\partial z}{\partial x^1}\partial_z +\frac{\partial \bar{z}}{\partial x^1}\partial_\bar{z} \Big\}T^1_{\ \ \ \nu}+\Big\{\frac{\partial z}{\partial x^2}\partial_z +\frac{\partial \bar{z}}{\partial x^2}\partial_\bar{z} \Big\} T^2_{\ \ \ \nu} \\ &=\partial_z\Big\{T^1_{\ \ \ \nu}+iT^2_{\ \ \ \nu}\Big\}+\partial_\bar{z}\Big\{T^1_{\ \ \ \nu}-iT^2_{\ \ \ \nu}\Big\} \end{align}
Since we are working in Euclidean metric I'm assuming indeces $\mu=1,2$ can be changed from upper to lower indistinctively (we can't do that if the indeces represent a $z$ or $\bar{z}$ since the metric is different in those coordinates). Using the energy-momentum tensor in complex coordinates
\begin{align} T_{zz}=\frac{1}{2}\big(T_{11}-iT_{12}\big) \\ T_{\bar{z}\bar{z}}=\frac{1}{2}\big(T_{11}+iT_{12}\big) \end{align}
Then I get $\partial_zT_{\bar{z}\bar{z}}+\partial_\bar{z}T_{zz}$ for $\nu=1$ and $-i\partial_zT_{\bar{z}\bar{z}}-i\partial_\bar{z}T_{zz}$ for $\nu=2$ (I'm using that the energy-momentum tensor is traceless symmetric to get these results). This two equations clearly don't resemble the two equations that Francesco gets, specially since he gets crossed terms in each equation. It seems like he is simply replacing the old indeces with complex indeces but I'd like to see how the equation is actually derived.
If anyone can point where I'm messing up the calculation it would be really helpful.