In this research study, we have proposed a new eighth-order optimal method. The new method consists of three steps: the Newton step, an optimal fourth-order iteration scheme, and a simply structured third step that improves the convergence order up to at least eight and ensures an efficiency index of 1.682. The discussion focused on the convergence analysis of the new method. We provide numerical examples to illustrate the effectiveness of our proposed method by comparing it to eighth-order numerical methods. Furthermore, we examine the complex dynamics and Attraction basins (AB), comparing them with various methods of the same order, and we present the results in summary tables.