Sum of square roots of complex numbers

Question

Is this always true?

$z^{1/2}+z^{1/2}=2z^{1/2}$

I said that it isn't in the multiform case but I want to justify this better (not just words).

Answer 1

For $z\neq 0$ we have $2$ distinct solutions

$$z^{\frac12}=w\implies w_1^2=z\quad w_2^2=z$$

and for any solution of course

$$w_i+w_i=2w_i$$

but

$$w_1+w_2=0$$