Extension to complex numbers

Question

Is there an extension to the complex numbers in which $zz^* = i$ has a solution? (The star denotes conjugation.)

EDIT:

I'm mathematically ignorant, but I'm guessing such an extension can't be a Banach algebra because norms in Banach algebras are non-negative reals.

Answer 1

I like to think that $\sqrt{-1}$ is really $\sqrt{-1*1}$ which is a sort of geometric mean of 1 and -1. So, the complex numbers enable us to calculate something which was inaccessible for certain numbers. (Of course, the geometric mean is also defined for more than two terms, e.g. the cube root of three numbers.)

An extension to complex numbers that would be useful:

• would help us be capable of calculations on things are currently "inaccessible"
• may not necessarily add an extra dimension to the complex plane
• etc.