From Hurwitz’s approach of complex continued fractions, we build a complex theory of the Pell’s equation. In this paper, we study the complex theory of the Pell’s equation,  that is, finding its solutions in Gaussian integers, using Hurwitz complex continued fraction, hence, generalizing it to the Pell’s equation  and in the process we also develop some interesting properties.

complex continued fraction, Hurwitz continued fraction, recurrence relation, Pell’s equation, Gaussian integers.