I've been working on solving some equations in the moduli and conjugate of complex numbers and it always seems to take me awfully long to arrive at an answer.
For example, solve $1+2|z|^2 = |z^2+1|^2+2|z+1|^2$. My first thought was to put $z=x+iy$ which didn't work out quite well since I got coupled quartic equations in $x$ and $y$. Then, I replaced $|z|^2=z \bar{z}$ and got an equation in $|z|,z, \bar{z}$ which I couldn't simplify directly either. Later, upon viewing the solution, the answer was reported to be $\omega^2$ i.e. the second cube root of unity, which, only in hindsight seemed obvious but I would not have seen it.
Sometimes, I've even noticed that problems use the triangle inequality or the geometry of complex numbers (like there was a problem which involved recognizing an equilateral triangle, from a system of 3 equations) in ways which are not so evident upon a first or second look. So, my question is what are the major techniques I can use to solve equations in the moduli and conjugates of complex numbers, and what should I always be on the lookout for to make an educated guess as to the form of the answer? Examples to demonstrate the use of these techniques would be appreciated!
Often, problems ask for the number of solutions to an equation or a pair of equations in complex numbers; for instance, find the number of common roots of $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$. Is there any way to answer these problems without explicitly finding the solutions? When confined to the real domain, it is usually easy to do so by sketching a rough graph but I can't do that with complex numbers.
Finally, if someone has a list of challenging or tricky problems solving equations of this kind in complex numbers or can direct me to such a list, I would be really thankful for the same.