4. Given that the complex number $z$ satisfies $z^{3}+z=2|z|^{2}$. Then all possible values of $z$ are $\qquad$
4. $0, 1, -1 \pm 2$ i.
$$
\begin{array}{l}
\text { Given } z^{3}+z=2|z|^{2}=2 z \bar{z} \\
\Rightarrow z\left(z^{2}+1-2 \bar{z}\right)=0 .
\end{array}
$$

When $z=0$, the condition is satisfied.
When $z \neq 0$, $z^{2}+1-2 \bar{z}=0$.
Let $z=a+b \mathrm{i}(a, b \in \mathbf{R})$. Then
$$
\begin{array}{l}
\left\{\begin{array}{l}
a^{2}-b^{2}+1-2 a=0, \\
2 a b+2 b=0
\end{array}\right. \\
\Rightarrow(a, b)=(1,0),(-1, \pm 2) .
\end{array}
$$

Therefore, all possible values of $z$ are $0, 1, -1 \pm 2$ i.