Sure. I have no idea what an efficient algorithm looks like, but since you only asked whether it's possible I'll offer a terrible one. Use Rouche's theorem to find $R$ such that all roots lie within a disk of radius $R$ centered at the origin, then subdivide the disk into, say, a mesh of squares of side length $\epsilon > 0$ and evaluate the polynomial at all the lattice points of the mesh. As the mesh size tends to zero you'll find points that approximate the zeroes to arbitrary accuracy.
Maybe a more efficient proposal is to use something like gradient descent on the square of the absolute value of the polynomial.