If $|z^2-1|=|z|^2+1$, how do we show that $z$ lies on imaginary axis ?
I understand that I can easily do this if I substitute $z=a+ib$. How do we solve it using algebra of complex numbers without the above substitution ?
My Attempt: $$ |z|^2+|1|^2=|z-1|^2+2\mathcal{Re}(z)=|z^2-1|\\ 2\mathcal{Re}(z)=|z^2-1|-|z-1|^2=|(z+1)(z-1)|-|z-1|.|z-1|\\=|z+1|.|z-1|-|z-1|.|z-1| $$ How do I proceed further and prove $\mathcal{Re}(z)=0$ ?