This problem has already been solved. In 2001
John Hubbard, Dierk Schleicher, and Scott Sutherland published a paper called "How to find all roots of complex polynomials by Newton's method" in Inventiones Mathematicae (one of the most prestigious math journals). I think the title pretty much speaks for itself.
Here's a short selection from a review of the paper by Peter Haïssinsky, which illuminates why this problem is difficult, and why (in my opinion) a naive approach to a solution is not likely to work.
"The paper under review answers the long-standing question of finding all the roots of any complex polynomial in one variable (by using Newton's method). This problem is difficult because, e.g., C. McMullen has proved that there are no generally convergent purely iterative algorithms for solving polynomials of degree 4 or more. Even in degree 3, there is a set of positive measure of polynomials for which a set of positive measure of initial guesses will not converge to any root with Newton's method.
"The authors answer this question by providing, for each degree d, a universal set S_d of initial guesses such that, for any polynomial of degree d (suitably normalized) and any of its roots, at least one point in S_d converges to it. This set has approximately 1.11d log_2 (d) points which are equally distributed on a small fraction of log d circles. In the case of real polynomials, this number can be brought down to 1.30 d. In the final section of the paper, the authors construct S_d explicitly."
Slashdot Mirror
Here's a short selection from a review of the paper by Peter Haïssinsky, which illuminates why this problem is difficult, and why (in my opinion) a naive approach to a solution is not likely to work.
"The paper under review answers the long-standing question of finding all the roots of any complex polynomial in one variable (by using Newton's method). This problem is difficult because, e.g., C. McMullen has proved that there are no generally convergent purely iterative algorithms for solving polynomials of degree 4 or more. Even in degree 3, there is a set of positive measure of polynomials for which a set of positive measure of initial guesses will not converge to any root with Newton's method.
"The authors answer this question by providing, for each degree d, a universal set S_d of initial guesses such that, for any polynomial of degree d (suitably normalized) and any of its roots, at least one point in S_d converges to it. This set has approximately 1.11d log_2 (d) points which are equally distributed on a small fraction of log d circles. In the case of real polynomials, this number can be brought down to 1.30 d. In the final section of the paper, the authors construct S_d explicitly."