Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$? I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & \hphantom{-}a\end{bmatrix}$.
But obviously $a+ib$ is quite different from $\begin{bmatrix}a & -b\\ b & \hphantom{-}a\end{bmatrix}$, as the latter being quite clumsy to use and seldom seen in any applications I am aware of. Furthermore, it complicates simple operations such as matrix multiplication whereby you have to go one extra step and extract the complex number after doing the multiplication.
Can someone explain what exactly is the difference (if there is any) between the two different representations? In what instances is a matrix representation advantageous?

Matrix representation of complex numbers is useful and advantageous because we can discover and explore new concepts, like this:
$\begin{bmatrix}\hphantom{-}a & b\\ -b & a\end{bmatrix}$ ---> complex numbers
$\begin{bmatrix}\hphantom{-}a & b\\ \hphantom{-}0 & a\end{bmatrix}$ ---> dual numbers
$\begin{bmatrix}\hphantom{-}a & b\\ \hphantom{-}b & a\end{bmatrix}$ ---> split complex numbers
