3. If $z \in \mathbf{C}, \arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, then the value of $z$ is
$$
\pm(1+\sqrt{3}) i
$$
3.【Analysis and Solution】As shown in the figure, using the geometric meaning of complex numbers, $z^{2}+4$ can be represented as $\overrightarrow{O A}$, $z_{2}-4$ can be represented as $\overrightarrow{O B}$, and $z^{2}$ can be represented as $\overrightarrow{O C}$. Since $\angle A O B=\frac{\pi}{2}$, and $C$ is the midpoint of segment $A B$,
$\because|O C|=4$, that is
$$
\begin{array}{l}
\left|z^{2}\right|=4, \arg z^{2}==\frac{5 \pi}{6}-\frac{\pi}{6}=\frac{2 \pi}{3} \\
\therefore z^{2}=4\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right), \\
\text { hence } z= \pm 2\left(\cos \frac{\pi}{3}+i \frac{\pi}{3}\right)= \pm 2\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)= \pm(1+\sqrt{3}) i .
\end{array}
$$