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?

Quirky little thing, there is a real version, little 2 by 2 blocks, of the Jordan Canonical Form that goes with this. For a real matrix, eigenvalues come in conjugate pairs; 
$$
\left(
\begin{array}{rrrr}
i & 1 & 0 & 0 \\
0 & i & 0 & 0 \\
0 & 0 & -i & 1 \\
0 & 0 & 0 & -i
\end{array}
\right)
$$
is similar to
$$
\left(
\begin{array}{rr|rr}
0 & 1 & 1 & 0 \\
-1 & 0 & 0 & 1 \\ \hline
0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0
\end{array}
\right)
$$
Then
$$
\left(
\begin{array}{rrrrrr}
i & 1 & 0 & 0 & 0 & 0 \\
0 & i & 1 & 0 & 0 & 0 \\
0 & 0 & i & 0 & 0 & 0 \\
0 & 0 & 0 & -i & 1 & 0 \\
0 & 0 & 0 & 0 & -i & 1 \\
0 & 0 & 0 & 0 & 0 & -i
\end{array}
\right)
$$
is similar to
$$
\left(
\begin{array}{rr|rr|rr}
0 & 1 & 1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 1 & 0 & 0 \\ \hline
0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & -1 & 0 & 0 & 1 \\ \hline
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & -1 & 0 \\
\end{array}
\right)
$$
