I know this question is vague or maybe broad and subjective.
But, I am interested in studying the equation $x^2+y^2=r^2$ when $x,y,r$ are complex numbers.
What are a few directions that I can follow to study this equation? Is it even interesting (subjective, I know)? Are there already results for this equation?
IMO it just represents that the squares of the complex numbers $x$ and $y$ addition through parallelogram law is the same as the square of the third complex number $r$.
If you take $x=a+ib, y=c+id, r= e+if$ then after simplifying a little bit you will get
$$a^2+c^2+f^2-b^2-d^2-e^2=0$$ $$ab+cd-ef=0$$ which doesn’t indicate anything geometrical to me at least.
You can calculate the argument of $r^2$ through $x$ and $y$ which is
$$\arg(r^2)=\arctan\left(\frac{2(ab+cd)}{a^2+c^2-b^2-d^2}\right)$$