Before the construction of the complex numbers, people thought you couldn't take the square root of a negative number. Then came along of the definition of the imaginary unit
$$i^2 = -1$$
and now we've got a whole new system with all-new properties and a few non-properties as well (like traditional ordering).
I was wondering if I could construct a unit that allows distance to be negative. Consider the definition
$$\left| j\right| = -1$$
Would this be possible? Could a whole new number system, with all-new properties (and perhaps non-properties) be constructed? What kinds of consequences would result from this definition?