Let $z_1 = p+qi$ and $z_2 = r+si$, then $|z_1+2z_2| = |2z_1+z_2| \implies |(p+2r) + i(q+2s)| = |(2p+r) + i(2q+s)|$ $\implies \sqrt{(p+2r)^2 + (q+2s)^2} = \sqrt{(2p+r)^2 + (2q+s)^2}$ $\implies r^2 + s^2 = p^2 + q^2 $ or equivalently, $|z_1| = |z_2|$. Therefore, we have that $(a^2-1)(r^2+s^2) = (a^2-1)(p^2+q^2)$ $\implies (p^2 +2apr+ a^2r^2) + (q^2 + 2asq + a^2s^2) = (a^2p^2 + 2apr + r^2 )+ (a^2q^2+2aqs+s^2)$ $\implies \sqrt{(p+ar)^2 + (q+as)^2} = \sqrt{(ap+r)^2 + (aq+s)^2}$ $\implies |z_1 + az_2| = |az_1+z_2|$.