3. In the complex number range, the number of common solutions of the equations $z^{4}+z=1$ and $|z|=1$ is $\qquad$ .
3.0.
Given $|z|=1$, we know that $z$ lies on the unit circle centered at $O$.
Furthermore, from $|1-z|=\left|z^{4}\right|=1$, we know that $z$ also lies on the unit circle centered at 1.
It is easy to see that the two circles intersect at two points, which are
$$
\begin{array}{l}
z_{1}=\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}, \\
z_{2}=\cos \left(-\frac{\pi}{3}\right)+\mathrm{i} \sin \left(-\frac{\pi}{3}\right) .
\end{array}
$$

From $z_{1}^{3}=-1 \Rightarrow z_{1}^{4}+z_{1}=-z_{1}+z_{1}=0$
$$
\Rightarrow z_{1}^{4}+z_{1} \neq 1 \text {. }
$$

Similarly, $z_{2}^{4}+z_{2} \neq 1$.
Therefore, the number of common solutions is 0.