Solve the equations:
a) $z^{4}=\bar{z}^{4}$;
b) $z^{2}+|z|=0$;
c) $z^{2}+\bar{z}=0$;
d) $z^{2}+|z|^{2}=0$;
e) $(z+i)^{4}=(z-i)^{4}$;
f) $z^{3}-\bar{z}=0$.
a) $a$, $ia$, $a(1 \pm i)$, where $a$ is an arbitrary real number;

b) $0, \pm i$; c) $0, -1, \frac{1}{2} \pm i \frac{\sqrt{3}}{2} ;$ d) $i a$; e) $0, \pm 1$; f) $0, \pm 1, \pm i$.