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 $|.|$. It is a property that must hold for all numbers, even if we expand our set.