1. Given the imaginary number $z$ satisfies the equation $z^{3}-3|z|-2=0$. Then $\left|z^{2}+4\right|=$ $\qquad$
,- 1.4 .
$$
\begin{array}{l}
\text { Given } z^{3}=3|z|+2 \Rightarrow|z|^{3}=3|z|+2 \\
\Rightarrow(|z|-2)\left(|z|^{2}+2|z|+1\right)=0 \\
\Rightarrow|z|=2 \text {. }
\end{array}
$$

Then $z^{3}=8 \Rightarrow(z-2)\left(z^{2}+2 z+4\right)=0$.
Note that, $z$ is an imaginary number.
Thus $z^{2}+2 z+4=0 \Rightarrow\left|z^{2}+4\right|=|-2 z|=4$.