Is Complex Analysis equivalent Real Analysis with $f:\mathbb R^2 \to \mathbb R^2$?
No, it isn't. This can most obviously be seen from the fact that the map $z \mapsto \overline z$ is not differentiable as a map from $\mathbb C$ to $\mathbb C$, but the corresponding map $(x,y) \mapsto (x,-y)$ is differentiable as a map from $\mathbb R^2$ to $\mathbb R^2$.
This is because to be complex differentiable you need to have a best complex linear approximation which is a much stronger requirement than to have a best real linear approximation.
Complex analysis is different from the analysis of functions $R^2 \to R^2$ in that it requires the functions not just to be differentiable, but complex-differentiable, which is a stricter requirement. One way of looking at the requirement is requiring that the functions locally look not only like a linear transformation $R^2 \to R^2$, but like a linear transformation corresponding to multiplication by a complex number, which we can identify with linear transformations of the plane by the obvious multiplication
$$(a,b) \cdot (c,d) = (ac - bd, ad + bc)$$
which is obvious when we write $(a,b) = a + bi$ and $(c,d) = c + di$. We can see this in matrix form if we identify $(a,b) = a+bi$ with the matrix
$$ \begin{bmatrix} a & -b \\ b & a \\ \end{bmatrix}.$$
Such functions have many properties that differentiable functions $R^2 \to R^2$ need not have, which gives complex analysis a much different flavor. For example, if we view complex functions on (simply connected subsets of) the complex plane as vector fields on $R^2$, then complex-differentiable functions will be conservative vector fields. However, if we know something about differentiable functions $R^2 \to R^2$, we then by definition know something about complex-differentiable functions $C \to C$, although we may need to do some translation.
The functions studied in complex analysis are much more specific than functions from $\Bbb{R}^2$ to itself.
If you like, you could say that complex analysis is the study of functions from $\Bbb{R}^2$ to itself which satisfy a particular system of PDE (namely the Cauchy-Riemann equations). Most people probably wouldn't phrase it that way, because it obscures the motivation; "we should extend the concept of differentiability to complex numbers" is a much more natural goal than "we should look at functions which satisfy some arbitrary PDE system."
Since there isn't a good 3-dimensional analogue of the Cauchy-Riemann equations, there also isn't a good 3-dimensional analogue of complex analysis. There are certain areas which extend, however; for example, the study of harmonic functions on $\Bbb{R}^2$ essentially reduces to complex analysis, but you can also study harmonic functions on $\Bbb{R}^n$ (and they have many of the same properties as their 2-dimensional counterparts).