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?

Bill Dubuques's answer shows how powerful matrix representations can be. For example using the same justification an element $a+b\sqrt2$ of the quadratic number field $\mathbf{Q}[\sqrt2]$ can be written as $1\quad 2b\choose 1\quad a$. In general elements of algebraic number field of degree $n$ can be represented as $n\times n$ matrices with rational numbers as entries. That is we have an injective  ring homomorphism from the number field $K$ to a (commutative) subring of non-singuler matrices over Q. The matrix of $\alpha$ being the matrix of the linear transformation corresponding to multiplication by $\alpha$. Norm and trace of the algebraic number are respectively the det and trace of the matrix.
