You can define complex numbers as pairs of real numbers with the product $$ (a,b)(c,d) = (ac - d\overline{b}, \; \overline{a}d + cb) $$ where \(\overline{a} = a\) for real numbers.

Then you can define define conjugation for complex numbers by $$ \overline{(a,b)} = (\overline{a}, -b) $$ Iterating this process, you can define quaternions as pairs of complex numbers...

... and octonions as pairs of quaternions!