Julia sets are fractal subsets of the complex plane defined by a simple iterative algorithm. Julia sets are specified by a single complex parameter and their appearances are indexed by the Mandelbrot set. This study presents a simple generalization of the quadratic Julia set that requires two complex parameters. The generalization causes the Mandelbrot set indexing the generalized Julia sets to become 4-dimensional and hence difficult to use as a visual index. An evolutionary algorithm is used to search the space of generalized quadratic Julia sets. A type of fitness function is presented that permits the artist exert some control over the appearance of the resulting Julia sets. The impact of different versions of the fitness function on the resulting Julia sets is explored. It is found that the designed fitness functions do give substantial control over the appearance of the resulting fractals.