I'd like to compare and contrast the derivative across different areas of mathematics just to organize ideas in my head. In this question, $f(x,y)$ means $f: \mathbb{R}^2 \to \mathbb{R}$ and $f(z)$ means $f: \mathbb{C}\to\mathbb{C}$
Limits
In multivariate calculus ($f(x,y)$), in order to find the limit of $f(x,y)$ as $(x,y) \rightarrow (x_0,y_0)$, you need to approach $(x_0, y_0)$ from all possible directions in the $xy$ plane. Likewise in complex analysis, in order to find the limit of $f(z)$ as $z \rightarrow z_0$, you need to approach $z_0$ along every path and check the limit.
Derivatives
In multivariate calculus, you usually talk about a derivative along a single direction. For instance, the partial derivative with respect to $x$ is given by
$$\frac{\partial f}{\partial x} = \lim_{h\to\ 0} \frac{f(x+h,y) - f(x,y)}{h}$$
This is the derivative parallel to the x-axis. You can generalize this to a derivative along any straight line direction. It's called a directional derivative. You can further generalize this to the derivative along any arbitrary path (not just straight lines) in your domain. However the two are sort of the same because at any one point on the arbitrary path, the derivative you compute there is a directional derivative at that instant of the path. Anyways, the point is to show that derivatives are taken with respect to directions.
For complex-valued functions of a complex variable, the derivative is
$$\frac{df(z)}{dz} = \lim_{h\to\ 0} \frac{f(z+h) - f(z)}{h}$$ with the understanding that h is a complex number that approaches $0$ along any path in the complex plane. So the derivative must be checked along every path and every path has to yield the same value of the limit in order for the derivative to exist.
Question
Why isn't there such notion of a derivative for multivariate functions $f(x,y)$? The domain of a function $f(z)$ is the complex plane. It's a plane. The domain of a function $f(x,y)$ is the $xy$ plane. Also a plane. Why can't I ask for the derivative of $f(x,y)$ at a point P where I check all possible paths as $(x,y) \rightarrow P$ just as I did for the derivative of $f(z)$? Why must I always take the derivative in a specific direction? Likewise, why can't you ask for the derivative of $f(z)$ in a certain direction at $z$?
My thoughts
Although a complex variable can take values in a complex plane, it's still a '1 dimensional number.' $z = x + iy$ is usually said to be a '2 dimensional number', but this is from the perspective of real numbers (1D) and generalizing to complex numbers (2D). However if I move the starting point in my brain to complex numbers, then a complex number is just a 1D number. Complex numbers live on a plane, but the plane should be thought of as 1D. Therefore just as checking the derivative of a single variable function $f(x)$ as $h$ goes to $0$ on 'all possible paths' is normal, checking the derivative of $f(z)$ as $h$ goes to $0$ on 'all possible paths' is normal because there aren't any paths (the plane is 1D). I can't ask for the $f'(z)$ along a particular direction because there is 'no direction' just as there is really no direction for a 1D real number line. However, I'm still confused why you can't ask for $f'(x,y)$ without specifying a direction. $f'(x,y)$ is always given with respect to a direction.