I know that:
1) A function $f:\mathbb{R}^2\to \mathbb{R}^2$, when differentiable at a point, has a $2\times 2$ matrix as a derivative, which is a linear transformation from $\mathbb{R}^2\to \mathbb{R}^2$ best approximating the function linearly in some neighbourhood.
2) There is a ring homomorphism $\mathbb{C} \to Mat_{2x2}(\mathbb{R})$ as $a+ib \longmapsto \left[\begin{array}{11}a & -b\\b & a \end{array}\right]$
3) For a function $f:\mathbb{C} \to \mathbb{C}$, I can define complex differentiabilty as the best $\mathbb{C}$-linear approximation of the function locally at a point, i.e., f'(z_0):h \mapsto f'(z_0)h
Now, I want to combine these three observations, so that the Cauchy-Riemann equation falls out by considering a complex differentiable function as a function from $\mathbb{R}^2\to \mathbb{R}^2$ and connect the jacobian with the $\mathbb{C}$-linear transformation via the homomorphism.
I am having trouble even formulating a proposition that I can prove. Do I define something called 'Complexfying an $\mathbb{R}^2$-operator'? Any help will be appreciated.
The upshot will be that I can then 'shift' the proofs of some of the basic results of holomorphic functions (such as the fact that if the partial derivatives of the co-ordinate functions exist and are continuous then the function will be holomorphic, etc) to that of multivariable calculus.