No. There is an important distinction between $i = \sqrt{-1}$, which is wrong, and $i^2 = -1$, which is correct, namely that the wrong relation implies that you can use square roots.
The main thing to keep in mind is that the square root is two-valued. This gives some weird behaviour, and it is best to stay away from it entirely when complex numbers are involved. Especially $\sqrt{a}\sqrt{b} = \sqrt{ab}$ fails, like you've just noted. It does stay true if we allow "flexible" choices of square roots, for instance
$$
1 = \sqrt{1} = \sqrt{(-1)\cdot (-1)} = \sqrt{-1}\cdot \sqrt{-1} = i \cdot (-i)
$$
where one of the square roots of $-1$ is chosen to be $i$, while the other is chosen to be $-i$. This is not a good habit to get into, and illustrates why you should at least be very careful with square roots when dealing with complex numbers.