In each case we have a quadratic form \(Q\) with $$ Q(x) = x\overline{x} \ge 0 $$ $$ Q(x) = 0 \iff x = 0 $$ and $$ Q(xy) = Q(x) Q(y) $$ So, the real numbers, complex numbers, quaternions and octonions are called
'normed division algebras'.

They're the only ones. Repeating the process does not give another!