Is there a generalization of the Rational Roots Theorem to complex rational roots? This answer on MSE shows that it can be applied to purely imaginary rational roots. I'm curious about roots of the form $\frac{m}{n}+\frac{p}{q}i$ however, where $m, p \in \mathbb{Z}$ and $n, q \in \mathbb{N^+}$.
Some Observations
Let's assume for simplicity that the polynomial $P(x)$ has only real-valued coefficients. Then if $\frac{m}{n}+\frac{p}{q}i$ is a root, so is $\frac{m}{n}-\frac{p}{q}i$ by the Complex Conjugate Root Theorem. This means that $P(x)$ is divisible by $x^2-\frac{2m}{n}x+(\frac{m^2}{n^2}+\frac{p^2}{q^2})$. Or in other words, $P(x)$ contains a quadratic factor with only rational coefficients, and where the leading term and constant term have the same sign.
Another observation is that we can rewrite $P(\frac{m}{n}+\frac{p}{q}i)$ as the sum of a real and imaginary polynomial over 4 variables, and we can set both of these polynomials equal to $0$. This approach gets out of hand really quickly however.
Any ideas on how to approach this? Does a generalized theorem simply not exist?