In this paper, we investigate some fundamental aspects of bi-complex numbers. Propose two different types of partial order relations and analyze bi-complex valued metric spaces under these orders, highlighting their differences. Moreover, we introduce the concept of a hyperbolic-valued metric space, examine the density of the natural numbers, and explore ideal convergence and the ideal Cauchy property for sequences of bi-complex-valued metric spaces. Finally, we discuss several properties within a bi-complex-valued metric space.
Hossain et al. (Sun,) studied this question.