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### Is there an impossible to solve equality in $\mathbb{C}$?

exactly like how ${x^2=-1}$ is impossible in $\mathbb{R}$ is there any equation that is impossible in $\mathbb{C}$ and how to deal with ?
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### Why not define $|v| = -1$? [closed]

I was wondering why if we have $i^2 = -1$, why not have a "number" $v$ such that $|v| = -1$? Does anything interesting arise from considering this system? The only thing I could come up with was: ...
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### Are complex numbers complete in every way?

I was told many times a story. Indeed a fascinating one to me as a student learning mathematics. First there were natural numbers. People started adding things and finding solutions to finding the ...
### Why there is no value for $x$ if $|x| = -1$? [duplicate]
According to the definition of absolute value negative values are forbidden. But what if I tried to solve a equation and the final result came like this: $|x|=-1$ One can say there is no value for $x$...