Square root notation and negative numbers

You can write $i = \sqrt{-1}$, but you have to take caution with it. If you extend the function $\sqrt{\cdot} \colon [0,\infty) \to \mathbf R$ somehow, say to some function $\sqrt{\cdot} \colon \mathbf R \to \mathbf C$, you cannot expect it to have the same properties as the function you started with, just because you happen to denote it with the same symbol. You correctly saw that any extension of $\sqrt{\cdot} \colon \mathbf R \to \mathbf C$ with $\sqrt{x}^2 = x$ for all $x\in \mathbf R$ will not have the property that $$ \sqrt{a}\sqrt{b}= \sqrt{ab} $$ Just because $$ \sqrt{-1}\sqrt{-1} = \sqrt{-1}^2 = -1 \ne 1 = \sqrt{(-1)(-1)} $$ Hence, you are expecting too much. And to prevent you from it, I'd rather not use the notation.