3. If $z \in C, \arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, then the value of $z$ is
3. 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 $\overrightarrow{O B}$, $z^{2}$ can be represented as $\overrightarrow{O C}$, and $\angle A O B=\frac{\pi}{2}$, and $C$ is the midpoint of segment $A B$,
$\therefore|O C|=4$, i.e.,
$$
\begin{aligned}
& \left|z^{2}\right|=4, \arg z^{2}=\frac{5 \pi}{6}-\frac{\pi}{6}=\frac{2 \pi}{3} . \\
\therefore \quad & z^{2}=4\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right),
\end{aligned}
$$

Therefore,
$$
\begin{aligned}
z & = \pm 2\left(\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}\right)= \pm 2\left(\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right) \\
& = \pm(1+\sqrt{3}) \mathrm{i} .
\end{aligned}
$$