Single variable complex analysis vs the world of the functions $f:\Bbb R^2 \to \Bbb R^2$. Is there any advantage to studying  single variable complex analysis as it is right now instead of just studying the world of the functions $f:\Bbb R^2 \to \Bbb R^2$?
I'm asking this because any function $\gamma: \Bbb C\to \Bbb C$ can be written as $\gamma(z)=u(x,y)+v(x,y)i$ and so we can associate the function $F: \Bbb R^2 \to \Bbb R^2, F(x,y)=(u(x,y),v(x,y))$ and study all the properties of a complex valued function as a real vector valued one. 
 A: It will help to start by writing down all the definitions.
What is a "real analytic" map $f: \Bbb R^2 \to \Bbb R^2$? It's a continuous function such that, in a small ball near a point $(x_0,y_0)$, $$f(x,y) = \sum_{n,m \geq 0} a_{n,m} (x-x_0)^n (y-y_0)^m.$$ A "complex analytic map" $f: \Bbb C \to \Bbb C$ is locally given by $f(z) = \sum a_n (z-z_0)^n$. More or less from this alone, one sees that $f(x,y) = x$ is real analytic but not complex analytic - its derivative at any point is not complex linear.
Let's go a little bit simpler. What does it mean for a map $\Bbb R^2 \to \Bbb R^2$ to be real differentiable? It means that near a point $(x_0,y_0)$, it has the form $f(x,y) = f(x_0,y_0) + A(x-x_0,y-y_0) + o(x^2+y^2)$, where $A$ is some linear map $\Bbb R^2 \to \Bbb R^2$. (If the partials exist and are continuous, then a map is real differentiable; the converse needn't be true.) On the other hand, it's much easier to say what we mean by complex differentiability: it means the limit $$\lim_{h \to 0} \frac{f(z+h)-f(z)}{h}$$ exists. Here $h$ is a complex number, not a real one, and division is the division of complex numbers. This is ultimately equivalent to saying that $f$ is real differentiable and $A$ is complex linear, and also equivalent to demanding the Cauchy-Riemann equations hold: if I decompose $f=u+iv$, where $u$ and $v$ are real-valued, I'm demanding $u_x=v_y, u_y=-v_x$.
The theorem you're thinking of says that if $f$ is complex differentiable, then actually it's complex analytic. There is no such result for real analyticity: $f(x,y) = x|x|$ is real differentiable but not real analytic at 0. You can think of the above result as being a remarkable "regularity" result for solutions to a certain kind of differential equation (the Cauchy-Riemann equations); much later in life you may learn that these are called elliptic equations, and this kind of regularity (where you automatically get smoothness, but here you even get analyticity) is called elliptic regularity. Further examples include the defining equation of harmonic functions: if $\Delta f = 0$ (meaning $f_{xx} + f_{yy} = 0$), and $f$ is $C^2$, say, then actually $f$ is smooth (even analytic!) on whatever open domain it's defined. This is a remarkable, beautiful, useful property, and complex analysis is the first place you can really see its power. 
A: The biggest obstacle in your path has more do with algebra than calculus. We can pretty easily multiply two complex numbers together in a way that is consistent with a field. This means that we can take two complex valued functions, multiply them, and end up with a complex valued function. The only way to do this with function $\mathbb R^2\to\mathbb R^2$ is to basically artificially introduce the same multiplication structure as in the complex numbers. To illustrate the importance of this, consider that most of the interesting and fundamental results from complex analysis come from integration: $$\int_\gamma f(z)\,dz.$$ If $z\in\mathbb R^2$ and $f(z)\in\mathbb R^2$, then how are we to multiply $f(z)$ and $dz$? Any way you come up with besides doing something equivalent to complex multiplication will not yield the same results as complex analysis.
