The Cauchy-Riemann conditions for the differentiability of $f(z) = f(x + iy) = u(x, y) + iv(x, y)$ in $(x_0, y_0)$ are
$$\displaystyle \frac{\partial u (x_0, y_0)}{\partial x} = \frac{\partial v (x_0, y_0)}{\partial y}$$
$$\displaystyle \frac{\partial u (x_0, y_0)}{\partial y} = - \frac{\partial v (x_0, y_0)}{\partial x}$$
These conditions are derived when $z = x + iy$ approaches $z_0 = x_0 + iy_0$ along the $x$-axis and the $y$-axis respectively. If the first derivative of $f(z)$ with respect to $z_0$ does exist, it must be not dependent on the pattern followed by $z$ approaching $z_0$.
So, how can the computation along just two possible patterns ($x$-axis and $y$-axis) be sufficient? How are (implicitly) considered all the other possible patterns while we obtain the Cauchy-Riemann conditions?
I know that they are sufficient conditions, but up to now, they appear to be not general and relative to only two specific cases of "approach" between infinite possibilities.