Complex numbers are said to be algebraically closed, meaning (to my mind) that given any polynomial x = p(z), with complex numbers x and z, the polynomial maps the complex number field back onto itself completely. For any given x, there will be a z. It is then stated that this makes any hyper-complex number superfluous for the analysis of arbitrary functions. Now it may be true that the hypercomplex number field is closed for any algebraic operation, and raising complex numbers to fixed powers. And most special functions also appear closed on investigation (I'm an engineer, not a mathematician, so hold your fire). What prevents me from defining an arbitrary function though, or finding one, that does not map the complex number field back onto itself completely? Some x = f(z), where for some x, there is not a z? finv(x) = z, z is not complex? If I then defined a hypercomplex number i2, and associated operational behavior whereby the field is once again complete, finv(x) = z, z is an element of hypercomplex field, then why is what I have just done fundamentally different than what was done to come up with complex numbers in the first place? After all, the original motivation for complex numbers was to define what happened when you inverted certain polynomials p(z) = x.