The Complex number a+bi may be defined as the matrix: $\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$

Addition, Subtract, Multiplication, and all further operations on these objects is precisely what Matrix Algebra (Linear Algebra) dictates they must be.

Example: What is |z|? From Linear Algebra: |z|² = zᵀ z = $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$ $\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$ = $\left[\begin{array}{cc}a\xb2+b\xb2& 0\\ 0& a\xb2+b\xb2\end{array}\right]$