TITLE: Could we invent a new number with $|p|=-1$?
QUESTION [9 upvotes]: We know that how a single definition $i^2=-1$ revolutionized our mathematics and solved many many problems. I wonder whether the definition $|p|=-1$ could have the potential of creating a new generation of numbers and help us in other areas like complex numbers do(from geometry to calculus). 
Has anyone researched on it?  
Please add appropriate tags.

REPLY [5 votes]: You can indeed modify the absolute value in $\Bbb Q$ to construct completions of it which are structurally very different from $\Bbb R$. This is how to define the $p$-adic numbers which depend on a preliminary choice of a prime number $p$. Nonetheless, even these "exotic" absolute values are always non-negative valued.