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. 
$$Re: \mathbb R^2 \to \mathbb R, \begin{bmatrix}
x\\
y
\end{bmatrix}\mapsto x \quad \text{and} \quad Im: \mathbb R^2 \to \mathbb R, \begin{bmatrix}
x\\
y
\end{bmatrix}\mapsto y$$

Yes, the space $\mathbb C$ is isomorphic to the $\mathbb R^2$ with respect to the corresponding maps (addition, multiplication etc.). 
It works like this: Consider the bijection
$$ \pi: \mathbb C \to \mathbb R^2, x+iy \mapsto \begin{bmatrix}
x\\
y
\end{bmatrix}$$
Now we define a fitting addition and multiplication on the $\mathbb R^2$. As addition we just take the entrywise addition, i.e.
$$\begin{bmatrix}
x_1\\
y_1
\end{bmatrix} + \begin{bmatrix}
x_2\\
y_2
\end{bmatrix} = \begin{bmatrix}
x_1 + x_2\\
y_1 + y_1
\end{bmatrix}$$
And we define a fitting multiplication, i.e.
$$\begin{bmatrix}
x_1\\
y_1
\end{bmatrix} \cdot \begin{bmatrix}
x_2\\
y_2
\end{bmatrix} = \begin{bmatrix}
x_1 x_2 - y_1y_2\\
x_1y_2 + x_2y_1
\end{bmatrix}$$
Now we can easily show that $\pi$ is a field isomorphism (left as exercise). Finally we can even define $Re$ and $Im$ like the following
$$Re: \mathbb R^2 \to \mathbb R, \begin{bmatrix}
x\\
y
\end{bmatrix}\mapsto x \quad \text{and} \quad Im: \mathbb R^2 \to \mathbb R, \begin{bmatrix}
x\\
y
\end{bmatrix}\mapsto y$$
This makes use of the fact, that $\mathbb C$ is a two dimensional $\mathbb R$-vector space (a base is for example $(1, i)$). 
Hope that helps you :)
