If you can show for two systems that whatever operations you carry out in one are always exactly mimicked in the other, then you can work in whichever system is the more convenient to use and all the results carry over to the other system. We say that the systems are isomorphic.

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.