9. (16 points) Given complex numbers $a_{1}, a_{2}, a_{3}$ satisfy
$$
\begin{array}{l}
a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=a_{1}^{3}+a_{2}^{3}+a_{3}^{3} \\
=a_{1}^{4}+a_{2}^{4}+a_{3}^{4}=0 .
\end{array}
$$

Find all possible values of $a_{1}+a_{2}+a_{3}$.
$$
\begin{array}{l}
\sum_{i=1}^{3} a_{i}^{3}=\left(\sum_{i=1}^{3} a_{i}\right)\left(\sum_{i=1}^{3} a_{i}^{3}\right)- \\
\left(\sum_{i \neq j} a_{i} a_{j}\right)\left(\sum_{i=1}^{3} a_{i}^{2}\right)+\left(\prod_{i=1}^{3} a_{i}\right)\left(\sum_{i=1}^{3} a_{i}\right) \text {. } \\
\text { By } \sum_{i=1}^{3} a_{i}^{2}=\sum_{i=1}^{3} a_{i}^{3}=\sum_{i=1}^{3} a_{i}^{4}=0 \\
\Rightarrow a_{1} a_{2} a_{3}\left(a_{1}+a_{2}+a_{3}\right)=0 \\
\Rightarrow a_{1}+a_{2}+a_{3}=0 \text { or } a_{1} a_{2} a_{3}=0 \text {. } \\
\end{array}
$$

If $a_{1} a_{2} a_{3}=0$, without loss of generality, let $a_{3}=0$.
Then the conditions become $\left\{\begin{array}{l}a_{1}^{2}+a_{2}^{2}=0, \\ a_{1}^{3}+a_{2}^{3}=0, \\ a_{1}^{4}+a_{2}^{4}=0 .\end{array}\right.$
Thus $a_{1}^{2} a_{2}^{2}=\frac{\left(a_{1}^{2}+a_{2}^{2}\right)^{2}-\left(a_{1}^{4}+a_{2}^{4}\right)}{2}=0$
$$
\begin{array}{l}
\Rightarrow a_{1} a_{2}=0 \Rightarrow a_{1}=0 \text { or } a_{2}=0 \\
\Rightarrow a_{1}=a_{2}=0=a_{3} \Rightarrow a_{1}+a_{2}+a_{3}=0 .
\end{array}
$$

In summary, $a_{1}+a_{2}+a_{3}=0$.