Complex z

Let $z=|z|e^{iα}$ . Then $|z|e^{iα}=\sqrt 2 |z|e^{i\frac{π}{4}}\implies |z|=0$

$\begin{aligned} z & = |z| + |z|i \\ & = |z|(1+i) \\ z(1-i) & = |z|(1+i)(1-i) \\ z - iz & = 2|z| & \small \blue{\text{Equating the imaginary part on both sides.}} \\ \implies z & = 0 \end{aligned}$

Therefore the sum of real part of all the solutions is $\boxed 0$ .