Studying dynamical systems is key to understanding a wide range of phenomena from planets' movement to climate patterns to market dynamics. Various numerical tools have been developed to address specific questions about dynamical systems, such as predicting the weather or planning the trajectory of a satellite. Considering the computability-theoretical aspects of dynamical systems gives rise to a myriad of intriguing directions. The focus of our study is dynamical systems that arise from iterating quadratic polynomials on the complex plane. They give rise to a variety of fractals known as Julia sets and are closely connected to the Mandelbrot set. In particular, while beautiful pictures of Julia sets are abundant on the Web, specific parameters can be constructed for which drawing the Julia set would require solving the Halting Problem.