finding all complex roots of equation

let $z = 1 +i$

Find all complex solutions such that $z^2 + \bar z^2 = 0$.

My working out:

$z^2 = -\bar z^2 = -(1-i)^2 = 2i$

so $z^2 = 2i$

hence $r^2 = 2 \implies r = \sqrt 2$

mod: $2\theta = \frac{\pi}{2} + 2k\pi \implies \theta = \frac{\pi}{4} + k(\pi)$ where $k = 0, 1$

overall roots are $z = \sqrt 2 \operatorname{cis} \left(\frac\pi4 +k\pi\right)$

Is my working out and solution correct?