I'm currently doing some reading on the Witt algebra, and I'm trying to understand the meaning behind $\frac{d}{dz}$.
In the book I'm reading (Conformal field theory by Martin Schottenloher), $\frac{d}{dz}$ is meant to represent a vector field in $\mathbb{C}$, that then defines a flow $\varphi_t(z)$ in $\mathbb{C}$, where each $\varphi_t$ is analytic. I've done a lot of searching for clarification on this notation, and I've found that often $\frac{d}{dz}$ is defined as $\frac{1}{2}\left(\frac{d}{dx}-i\frac{d}{dy}\right)$, where $z = x+iy$. However it is still not clear to me what it means to flow according to $\frac{1}{2}\left(\frac{d}{dx}-i\frac{d}{dy}\right)$ in $\mathbb{C}$.
Is there a standard book on Complex manifolds that perhaps goes into detail about flows defined by complex vector fields?