We improve the method of
Janovská and Opfer for computing the zeros on the surface of a given sphere for a
quaternionic two-sided polynomial. We classify the zeros of quaternionic two-sided
polynomials into three types —isolated, spherical and circular—and characterize each
type. We provide a method to find all quaternion zeros for two-sided polynomials
with complex coefficients. We also establish standard formulae for roots of a
quadratic two-sided polynomial with complex coefficients, which yields a
simpler and more efficient algorithm to produce all zeros in the quadratic
case.