Could we invent a new number with $|p|=-1$?

There is a difference here.

The value $i$ is defined as the number that solves the equation $x^2+1=0$. The reason that this equation does not have a solution in $\mathbb R$ is that for every $x\in\mathbb R$, $x^2>0,$ which is a consequence of the properties of the real numbers. There is nothing inherit in the equation that would demand it to have no solution.

On the other hand, the value $|x|$ is defined to always be positive, thus it will by definition never equal to $-1$.

The difference then:

• $x^2>0$ is a consequence of the properties of real numbers. Looking at numbers with different properties may change this fact.
• $|x|>0$ is inherit in the definition of $|\cdot|$. It is a property that must hold for all numbers, even if we expand our set.

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.