How can $ i $ be distinguished from $ - i $?

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Mathematicians designate one solution to $x^2 = -1$ as $i$ and the other as $-i$. Would anybody notice if we switched their identities? Any polynomial $p(x)$ with a complex root will also have its conjugate as a root. Is there any equation in real numbers with a complex solution that does not have its conjugate as another solution?