Why can't we define "the" derivative when mapping from $\mathbb{R}^2 \to \mathbb{R}$? When we map from $\mathbb{R} \to \mathbb{R}$, the derivative is given by $\lim_{x  \to a} \dfrac {f(x) - f(a)}{x-a}.$ When we map from $\mathbb{C} \to \mathbb{C}$ (basically $\mathbb{R}^2 \to \mathbb{R}^2$) we also have a definition for the derivative, $\lim_{z  \to w} \dfrac {f(z) - f(w)}{z-w}.$
However, when we map  $\mathbb{R}^2 \to \mathbb{R}$, we have only partial derivatives, $\dfrac {\partial f}{\partial x}$, $\dfrac {\partial f}{\partial y}$, and directional derivatives. Why is it that we can't define the derivative just as we do in the other $2$ cases? Is it because there is no nice way to define division between a member of $\mathbb{R}$ and a member if $\mathbb{R}^2$?

Because $\mathbb{C} \to \mathbb{C}$ is not ''basically the same'' as $\mathbb{R}^2 \to \mathbb{R^2}$.
The possibility of define a derivative for a function $ f:\mathbb{C} \to \mathbb{C}$ use , as you noted, the fact that $ \mathbb{C}$ is a field, and this implies that the derivative, if it exists, is much more than the linear approximation of the function near a point.
