Writing Complex Numbers as a Vector in $\mathbb{R^2}$ Generally speaking is it mathematically correct to write a complex number $z = x +  iy \in \mathbb{C}$, as a vector in $\mathbb{R}^2$?
$$z=\begin{bmatrix}
x\\
y
\end{bmatrix} \quad \text{with } x,y \in \mathbb{R}$$
With row 1 being the $Re(z)$ and row 2 being $Im(z)$, i.e. 
$$Re: \mathbb R^2 \to \mathbb R, \begin{bmatrix}
x\\
y
\end{bmatrix}\mapsto x \quad \text{and} \quad Im: \mathbb R^2 \to \mathbb R, \begin{bmatrix}
x\\
y
\end{bmatrix}\mapsto y$$

Because the complex number space $\mathbb C$ is isomorphic to $\mathbb R^2$, the answer is yes. However, writing $(a+bi) \cdot (c+di)$ as
$$\begin{bmatrix}
a\\
b
\end{bmatrix} \cdot 
\begin{bmatrix}
c\\
d
\end{bmatrix}$$
does not make sense under normal vector multiplication - so you will need to define the multiplication operator in a different way.
Another way of doing this is to write complex numbers $a+bi$ as the matrix
$$\begin{bmatrix}
a&-b\\
b&a
\end{bmatrix}$$
which has the advantage of normal matrix-matrix multiplication giving the same result as normal complex multiplication.
