complex numbers - Simplify $sqrt{-3}$

I was reading about this known fallacy
$$
-1 = i^2 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1
$$
and according to Wikipedia "The fallacy is that the rule $\sqrt{xy} = \sqrt{x}\sqrt{y} $ is generally valid only if both x and y are positive"

So my question is, how come we can say that $\sqrt{-3} = \sqrt{3}i$ ?. Aren't we applying the same mistake as the fallacy? Like $\sqrt{-3} = \sqrt{(-1)(3)} = \sqrt{-1}\sqrt{3} = \sqrt{3}i$ cannot be since -1 is negative.

Thanks for reading.