Consider the unital commutative algebra defined by
A^2 = B
B^2 = A
AB = BA = -1
so we get numbers of the form x + y A + z B where x,y,z are real.
I call it the anticomplex numbers or anticomplex algebra.
It is one of the most interesting 3D numbers in my opinion.
Notice it is not possible to "solve" it in terms of complex numbers ; you cannot find a solution for A and B that are complex numbers.
It is also a non-associative algebra.
And , just as with complex numbers , there is no nonzero number S such that S^2 = 0.
It turns out that many 3D numbers are isomorphic to the anticomplex numbers.
And in the classification of 3D numbers they are important.
There do exist zero-divisors in this algebra.
Thank you for your interest.
tommy1729