Akiyama et al. (2008) introduced an algorithm to find the expansion of a positive integer in a rational base a/b, which is different from the greedy algorithm. We will introduce a generalized version of this algorithm to expand complex numbers in Gaussian rational bases of modulus greater than one. We will give an overview of the shape of these expansions and characterize them in terms of p-adic completions with respect to Gaussian primes, as well as relate them to fractal sets. If time permits, we will tackle the questions of finiteness property and uniform distribution of digits.