Since you mentioned this, here's another approach:

Take absolute values of both sides of

, getting

, so |z|=0 or |z|=1. Since 0 is easily seen to be a solution, we're focused on possible solutions on the unit circle.

Now multiply your original equation by z, getting

. Do you know an identity relating

and |z|? This should lead you to consider cube roots of unity, which have a nice geometric interpretation. You might get some extraneous solutions here though (possibly, too tired to trust myself now lol).