The intricately interwoven basins of attraction stemming from Newton's Method applied to a simple complex polynomial are a common sight in fractal, dynamical systems, and numerical analysis literature. In this work, the author investigates how this workhorse of root-finding algorithms works for complex polynomials, in addition to a variety of other settings, from the simple, one-dimensional real function with a simple root, to the infinite-dimension Banach space. The rapid, quadratic convergence of Newton's method to a simple root is well known, but this performance is not guaranteed for all roots and for all starting points. Damping is one modification to the Newton algorithm that can be used to overcome difficulties in global convergence. We explore computationally how this damping affects the fractal geometry of the Newton basins of attraction for a simple function.