I will discuss integral solutions of the Markov equation and the unicity conjecture. I will discuss a few ways of parameterizing Markov triples using modular arithmetic and factorizations of Markov and related numbers over the Gaussian integers, giving rise to factorizations of the Markov equation over the Gaussian integers. I will interpret these factorizations geometrically in terms of curves on the modular torus and their lifts to the hyperbolic plane.