Example 6 Given a complex number $z$ satisfying $|z|=1$, try to find the maximum value of $u=\left|z^{3}-3 z+2\right|$.
(2000 Jilin Province High School Competition)
$$
\begin{aligned}
u & =\left|\left(z^{3}-z\right)-2(z-1)\right| \\
& =\left|(z-1)\left(z^{2}+z\right)-2(z-1)\right| \\
& =\left|(z-1)^{2}(z+2)\right| .
\end{aligned}
$$

Let $z=x+y i, x, y \in \mathbf{R}$, by $|z|=1$ we get $x^{2}+y^{2}=1$, and $|x| \leqslant 1$, then
$$
\begin{aligned}
u & =\sqrt{\left[(x-1)^{2}+y^{2}\right]^{2}\left[(x+2)^{2} y^{2}\right]} \\
& =\sqrt{(-2 x+2)^{2}(4 x+5)} \\
& =\sqrt{(2-2 x)(2-2 x)(4 x+5)} \\
& \leqslant \sqrt{\left(\frac{(2-2 x)+(2-2 x)+(4 x+5)}{3}\right)^{3}} \\
& =3 \sqrt{3} .
\end{aligned}
$$

When and only when $2-2 x=4 x+5, x=-\frac{1}{2}$, i.e., $z=-\frac{1}{2} \pm \frac{\sqrt{3}}{2} \mathrm{i}$, $\max u$ $=3 \sqrt{3}$.