This formula for nested-radicals: $\sqrt{A+\sqrt{B}}=\sqrt{\frac{A+C}{2}}+\sqrt{\frac{A-C}{2}}$, works fine with $B > 0$, and with $A + \sqrt{B} > 0$, but doesn't work with complex numbers.
I tried to do a demonstration for imaginary numbers, but it did not work:
$\sqrt{A + \sqrt{B}i} = \sqrt{x} + \sqrt{y}i$
$A + \sqrt{B}i = x + \sqrt{4xy}i -y$
$A + \sqrt{B}i = x - y + \sqrt{4xy}i$
$A = x-y, \sqrt{B}i = \sqrt{4xy}i \implies B = 4xy \implies x = A +y$
$B = 4(A+y)y \implies 4Ay +4y^2 \implies 4y^2 + 4Ay - B = 0$
$y = \dfrac{-4A \pm \sqrt{16A^2+16B}}{8} \implies \dfrac{-A\pm\sqrt{A^2+B}}{2}$, but this not work.