Generalizations of complex numbers

Quaternions follow the rule ``i²=j²=k²=ijk=-1'' (which must suck for people who don't know the ordering of the English alphabet). From this we can derive ij=k, jk=i, ki=j, and reversal negates: ji=-k kj=-i ik=-j. from which we can get the squaring rule (a+bi+cj+dk)² = (a²-b²-c²-d²)+2abi+2acj+2adk Squaring is interesting because z:=z²+c yields fractals. Next idea: complex numbers of complex numbers, where we recursively apply the "square the second component to get the negative of the first component" rule. The normal way to write it out would be with coordinates, but I'll try ijk notation and see what happens: ((a,b),(c,d))²= (a+bi+cj+dk)²= (a²-b²-c²+d²)+(2ab-2cd)i+(2ac-2bd)j+(2bc+2ad)k The multiplication table looks like

    1  i  j  k
  * ----------
1 | 1  i  j  k
i | i -1  k -j
j | j  k -1 -i
k | k -j -i -1

Generalizing complex numbers to 3-vectors of the form (a,(b,c)) runs into the problem of how to define the result of X+(Y,Z) to be a scalar.