Derivative and partial derivative of complex functions

The relation you observe is exactly how we get to the Cauchy-Riemann equations for the real and imaginary parts of an analytic function.

The complex derivative of a function $f:U\to{\bf C}$ at $z_0\in U$ where $U$ is an open subset of ${\bf C}$ is defined by $$ f'(z_0)=\lim_{z\to z_0:z\in U\backslash\{z_0\}}\frac{f(z)-f(z_0)}{z-z_0}\tag{1} $$ If ${f}$ is complex differentiable at ${z_0}$, then by specialising the limit (1) to variables ${z}$ of the form ${z = z_0 + h}$ for some non-zero real ${h}$ near zero we have $$ \lim_{z\to z_0:z\in U\backslash\{z_0\}}\frac{f(z)-f(z_0)}{z-z_0} =\lim_{h\to 0:h\in{\bf R}\backslash\{0\}}\frac{f((x_0+h)+iy_0)-f(x_0+iy_0)}{h} =u_x(z_0)+v_x(z_0)=:\frac{\partial f}{\partial x}(z_0) $$ where $z_0=x_0+iy_0$ and $f=u+iv$.

The functions you consider are all analytic, therefore your $x$-derivative will coincide with the complex derivative.

Analyticity ensures that the complex derivative coincides with the 'partial' derivatives with respect to the real/imaginary axis. Formally, analyticity means that the complex limit $$\lim_{z\to 0} \frac{f(z_0 + z) - f(z_0)}z$$ is independent of the way that $z\to 0$. In particular you could just take the limit along the real axis (which would just correspond to your $x$-derivative) and obtain the same result. But you could also take the limit along the imaginary axis, and should obtain the same result as before. Analyticity ensures that. The property that those limits are the same is ensured by the Cauchy-Riemann equations, which are equivalent to analyticity (for a continuous function on an open set).

If a function is not analytic, the complex derivative need not be well defined, and the direction in which you take your derivative might influence the outcome. An analogy is the question whether a function $f:\mathbb R^2\to\mathbb R$ is Frechet-differentiable or not. Hence, for a non-analytic function, the complex derivative will not exist, and it plays a role whether you take the $x$-derivative or the $\mathrm{i}y$-derivative.

There is a difference between a general function of two variables f(x,y) and a complex function f(z)=f(x+iy). In the first, the function variations with x and y are completely independent, whereas in a complex function the variable (x+iy) transforms wholly. Accordintly, whatever happens to x, happens to y too (and hence z as well) in terms of the slope, which is the (partial) derivative. Hope this helps.

Masoud