Julia sets for complex-valued polynomials have been well-studied for years. However, the graphs of the polynomials themselves and their iterates are more difficult to visualize because they are four-dimensional. In this paper, we explore the dynamics of these functions by analyzing the behavior of the real and imaginary parts of the iterates. We also define two sets of points for which the real (respectively imaginary) parts of the iterates remain bounded, and prove how these sets relate to the corresponding filled Julia set. We end by applying our results to the well-known class of functions f_c(z) =z²+c.