15. In the complex plane, the set
$$
\begin{array}{l}
A=\left\{z \in \mathbf{C} \mid z^{3}-8=0\right\}, \\
B=\left\{z \in \mathbf{C} \mid z^{3}-8 z^{2}-8 z+64=0\right\} .
\end{array}
$$

Then the maximum distance between points in sets $A$ and $B$ in the complex plane is ( ).
(A) $2 \sqrt{3}$
(B) 6
(C) 9
(D) $2 \sqrt{21}$
(E) $9+\sqrt{3}$
15. D.

From the problem, we know
$$
\begin{array}{l}
A=\left\{z \in \mathbf{C} \mid z^{3}-8=0\right\} \\
=\{(2,0),(-1, \sqrt{3}),(-1,-\sqrt{3})\}, \\
B=\left\{z \in \mathbf{C} \mid z^{3}-8 z^{2}-8 z+64=0\right\} \\
=\{(8,0),(-2 \sqrt{2}, 0),(2 \sqrt{2}, 0)\} .
\end{array}
$$

Therefore, $d_{\max }=\sqrt{(8+1)^{2}+(0 \pm \sqrt{3})^{2}}$
$$
=\sqrt{84}=2 \sqrt{21} \text {. }
$$