Yes. Write a complex number $a+bi$ in polar form $ce^{it}$. Then
$$a^2+b^2=(a+bi)(a-bi)=(ce^{it})(ce^{-it})=c^2.$$
This is the first theorem I prove in my complex analysis class (after defining complex multiplication via the polar form and checking that it agrees with the more traditional definition via $i^2=-1$ and distributivity).