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.
There are also lots of specialized algorithms for finding roots of polynomials at the Wikipedia article.