In real algebra, if I have a differentiable function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, say $f(x,y)=[u,v]$, I can calculate four different partial derivatives $\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y}$. If I interpret $x$ and $y$ as spatial coordinates, I could - for example - create two separate quiver plots: one illustrating the partial derivatives of $u$ and one illustrating the partial derivatives of $v$.
In complex algebra, it seems like we should encounter a similar behaviour: Assume that I have $z=x+iy$ and $w=u+iv$, and that my function $f:\mathbb{C}\rightarrow \mathbb{C}$ is (say) a Mobius transformation
$$M(z)=\frac{az+b}{cz+d}$$
whose derivative is:
$$\frac{\partial w}{\partial z}=\frac{\partial M(z)}{\partial z}=\frac{ad-bc}{(cz+d)^2}$$
Now here is my question: The derivative I obtain will be a single complex number. Is it possible to extract the partial derivatives $\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y}$? If yes, how would I go about this?
Yes, this is possible. In general we have
$$\frac{\mathrm df}{\mathrm dz}=\frac{\partial u}{\partial x}+\mathrm i\frac{\partial v}{\partial x},$$
and the Cauchy-Riemann equations say that in addition,
$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\\ \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.$$
Together, this allows us to extract all the partial derivatives from $\frac{\mathrm df}{\mathrm dz}$ if we know its real and imaginary parts.
In this question you can find some ideas about how to obtain these equations from the definition of real and complex differentiability.