Using a set of matrices exhibits all the algebraic structure of complex numbers including a matrix with real entries that corresponds to $\sqrt -1$. Having established the model it is more convenient to use the $x+i y$ notation rather than use the matrices.
Using a set of linear combinations of matrices exhibits all the algebraic structure of quaternions including three different matrices corresponding to the three different square roots of -1. Again, having established the model, it is more convenient to use the $a + {\bf i}x + {\bf j}y + {\bf k}z$ notation rather than to use the matrices.