Complex derivative as a real derivative? Part (a) 
Suppose $f'(z)$ is a complex derivation of $f(z)$. Since $f'(z)$ takes complex values, $f'(z)$ is a $2 \times 1$ column vector in $\mathbb{R}^2$. 
Part (b)
If I interpret $f$ as function between $\mathbb{R}^2 \rightarrow \mathbb{R}^2$, then the derivation $f'(z)$ becomes a $2 \times 2$ linear tranformation.
Conclusion of Part (a) and Part (b) don't match up. Why?
 A: The real derivative of a holomorphic function (which is required for the complex derivative to exist) at a point is the product of some multiple of the identity matrix with a rotation matrix (this is the reason for the name "holomorphic": it preserves angles, unless the derivative is $0$). Thus only a $2$-dimensional subspace of the $2\times2$ matrices appear. Specifically, only matrices of the form
$$
\begin{bmatrix}a&-b\\b&a\end{bmatrix}
$$
for some real $a,b$ appear in this setting. Alternatively, the Cauchy-Riemann equations are satisfied iff the real derivative has exactly this form.
The corresponding complex derivative is $f'(z)=a+bi$, or $$\begin{bmatrix}a\\b\end{bmatrix}$$in your matrix form.
A: They don't match up because they are different objects. This should be familiar: for functions $f:\mathbb R\to \mathbb R,$ $f'(x)$ is a number, while $Df(x)$ is the unique linear transformation from $\mathbb R$ to $\mathbb R$ such that $f(x+h)-f(x) = Df(x)(h) + o(h).$ We do have $Df(x)(h) = f'(x)h$ for all $h,$ so the two objects have a direct connection, but they shouldn't be confused with each other, and therefore the same notation, $f'(x),$ shouldn't be used for both of them - at least not until the difference between the two is fully absorbed.
Moving to $\mathbb C= \mathbb R^2,$ things will again be confusing if the notation $f'(z)$ is used for two different objects, namely the complex derivative, which is the number $f'(z),$ and the corresponding (real) linear transformation $Df(z):\mathbb R^2\to \mathbb R^2.$ It's the same "problem" as in the first paragraph. There's no actual problem unless we get careless with the notation.
