definition - Not defining the imaginary number $i$ as the principal square root of $-1$.

Background

I learned early on that it's important that we define the imaginary number $i$ such that $i^2 = -1$, rather than $i = \sqrt{-1}$.

Question

I can't fully remember the reasoning for this important note, so I was wondering if someone could elaborate?

Own efforts

Any explanation I find eventually boils down to the same argument.

If we define $i$ as the principal square root of $-1$, then we get

$$-1 = i^2 = \sqrt{-1}\sqrt{-1} \overbrace{=}^{\text{fallacy}} \sqrt{(-1)(-1)} = \sqrt1 = 1$$

But to me, this seems like wrongful use of the $\sqrt{ab} = \sqrt a \sqrt b$ rule, since this rule comes with certain restrictions on $a, b$. So I don't see how this is a misuse of the definition of $i$.

Are there other reasons why we should be careful not to define $i$ as the principal square root of $-1$?

Answer

If you define $i$ as $\sqrt{-1}$ then there is an obvious question: how do you know that $-1$ has some square root? Besides, writing $i=\sqrt{-1}$ seems to imply that $i$ is the square root of $-1$. But, in $\mathbb C$, $-1$ has two square roots: $\pm i$. Assuming that $i$ is the square root of $-1$ leads to fallacies, such as the one the you mentioned.