Notice that the mapping $\|\cdot\|_* \colon \mathbb{C}^n \to \mathbb{R}$ given by $\|(a_1,\dots,a_n)\|_* = \sup_{z_1^3=z_2^3=\dots=1} |a_1z_1 + \dots + a_n z_n|$ defines a norm. The result then follows by the equivalence of norms in finite dimensional vector spaces.
The triangle inequality is easily seen to be true: Just use the ordinary triangle inequality. To prove that if $\|(a_1,\dots,a_n)\|_* = 0$ then $(a_1,\dots,a_n) = 0$ notice that if one of the coordinates (let's say $a_1$) wasn't $0$, then there would be three different numbers: $a_1 z_1 + a_2 + \dots + a_n$, $a_1 z_2 + a_2 + \dots + a_n$ and $a_1 z_3 + a_2 + \dots + a_n$ one of which is non-zero and thus the sup we get would be non-zero also.