When I was 17 I tried hard to find a similar construction for complex numbers in 3d (in order to make fractals) and ´discovered´ the quaternions. I still think the way I devised them is more natural: It needs only two operators I^2=-1,J^2=-1, and the rule I J=-J I so, all multiplication for a +b I + c J + d I J follows. If you add another operator you get the octonions and so on.
When I was 17 I tried hard to find a similar construction for complex numbers in 3d (in order to make fractals) and ´discovered´ the quaternions. I still think the way I devised them is more natural: It needs only two operators I^2=-1,J^2=-1, and the rule I J=-J I so, all multiplication for a +b I + c J + d I J follows. If you add another operator you get the octonions and so on.