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).