optimization: feasible algorithm to find interpolation of complex polynomials based on absolute values

Consider a complex degree$ (n-1) $ polynomial $ p (z) = sum limits_ {i = 0} ^ {n-1} a_i z ^ i $. Given a number $ 0> m> 2n $ of positions in the complex plane with absolute value requirements, that is, $ | p (z_j) | overset {!} {=} b_j $ (with $ b_j geq 0 $), there is a numerically stable and feasible algorithm to find the coefficients $ a_i $ such that $ p (z) $ satisfies those requirements? How big can $ m $ be for such an algorithm?

In other words, is there a way to solve a complex polynomial interpolation problem based only on given absolute values, leaving the argument (angle) of the polynomial completely arbitrary at any point?