The number of complex numbers z satisfying z^4=i bar z and Re(z)≥ 0 is

Taking moduli, $|z|^4=|z|$, so $z=0$ or $|z|=1$. For $|z|=1$ we have $\bar z=1/z$, hence $z^5=i$ and $z=e^{i\pi(1+4k)/10}$, $k=0,1,2,3,4$, with arguments $18^\circ,90^\circ,162^\circ,234^\circ,306^\circ$. The real part is non-negative for $18^\circ$, $90^\circ$ (real part $0$) and $306^\circ$, while $162^\circ$ and $234^\circ$ have negative real part. That gives three unit-modulus solutions. The root $z=0$ also satisfies the equation and has $\operatorname{Re}z=0\ge0$, so it must be counted. Total $3+1=4$. Choosing 3 forgets $z=0$; choosing 6 ignores the half-plane restriction.