We develop the escape criterion for a complex-valued cosine function \(T(z) = \cos (z^n)+bz+c\) , where \(b,c \in \mathbb {C}\) , and \(n \ge 2\) , using four distinct iterations. We use our results to explore mutants of the classical Julia sets in Noor, Ishikawa, Mann, and Picard orbits and observe certain patterns in these distinct orbits. We notice that most of the visualized fractals are similar to beautiful natural objects and the Picard fixed-point iteration does not give Julia sets with expected characteristics (shape, size, color, and resolution). This leads to a topic of future research to find the reason for failing the Picard iteration in generating fractals as obtained by the other three fixed-point iterations. Since Picard-generated Julia sets are highly complex and when we zoom in, we uncover hidden complex shapes. The motivation behind this work is the fact that numerous objects existing in nature have a property of self-similarity. For example, trees, seashells, mountains, the nervous system of human beings, and so on. Sometimes this feature is very prominent. Identification of these natural structures is a challenging problem in computer vision. These are not actually identical to each other, however, these are similar having similar characteristics, but not exactly the same.