  We propose the extension of the complex numbers to be the new domain where
new concepts, like negative and imaginary probabilities, can be defined. The
unit of the new space is defined as the solution of the unsolvable equation in
the complex domain: $|z|^2= z^* z = i$. The existence of the unsolvable
equation in a closed domain as complex's lead to the definition of a new type
of multiplication, for not violate the fundamental theorem of algebra. The
definition of the new space also requests the inclusion of a new mapping
operation, so the absolute value of the new extended number being real and
positive. We study the properties of the vector space like
positive-definiteness, linearity, and conjugated symmetry.
