How to compute the norm of a complex number under square root?

How to compute the norm of a complex number under square root? Does the square of norm equal the norm of square:

$\|\sqrt z\|^2 = \|\sqrt {z^2}\|$?

Let $z = re^{i\theta}$, then $$\|\sqrt z\|^2 =\|\sqrt {re^{i\theta}}\|^2 = \|\sqrt r \sqrt {e^{i\theta}}\|^2 =\|\sqrt r {e^{1/2i\theta}}\|^2 = \|r {e^{i\theta}}\|.$$
And
$$\|\sqrt {z^2}\|=\|\sqrt {(re^{i\theta})^2}\| = \|\sqrt {r^2e^{2i\theta}}\|= \|{re^{i\theta}}\|.$$

I hope this is correct? Thank you.