This question is related to: What's the correct way of defining the use of square root symbol?
As far as I know, the radical symbol $\sqrt{}$ only denotes the principal square root, even in the complex field. So $\sqrt{-4}$ should only be equal to $2i$ and not to $-2i$, because $-2i$ is the secondary square root $-\sqrt{-4}$. However, if we use $-4$ in the De Moivre's formula, we can get both answers.
In the same way, putting $4$ in that formula brings the two answers $2$ and $-2$, so the same expression $\sqrt{4}$ could behave differently depending on the field in which we are working.
But I thought that maybe the mistake comes because de Moivre's formula is not intended to be used for fixed values of $z$, but to find the solutions of $z^n = Expression$.
This in the same sense that in reals the equation $\sqrt{x^2} = \sqrt{25}$ (that comes from $x^2 = 25$) has multiple solutions because in that case we are asking for the inputs of the function $\sqrt{x^2}$ that provide the given output, while in the case of $\sqrt{5^2} = x$ the solution is only one because we already have the input and we are asking for the output.
Thus, the use of de Moivre's formula for $\sqrt[2]{z}$ could be for $z$ unknown, similar to the case of $\sqrt{x^2}$.