4. In the complex plane, the number of intersection points between the curve $z^{4}+z=1$ and the circle $|z|=1$ is ( ).
(A)0
(B) 1
(C) 2
(D) 3
4. (A).

Let $z=\cos \theta+\mathrm{i} \sin \theta(0 \leqslant \theta<2 \pi)$, then $z^{3}+1=\bar{z}$. Therefore, $1+\cos 3 \theta+i \sin 3 \theta=\cos \theta-i \sin \theta$, which means $\left\{\begin{array}{l}\cos 3 \theta=\cos \theta-1, \\ \sin 3 \theta=-\sin \theta\end{array}\right.$
$$
\therefore(\cos \theta-1)^{2}+(-\sin \theta)^{2}=1 \text {. }
$$

Thus, $\cos \theta=\frac{1}{2}$.
Then $\theta=\frac{\pi}{3}$ or $\frac{5}{3} \pi$ do not satisfy $1+\cos 3 \theta=\cos \theta$.