If the category in question is concrete with underlying set functor $C \to Set$, and bimorphisms are given by one-to-one and onto functions, then the more interesting question is what happens _inside_ a bimorphism class. Namely, what sort of $C$-objects are there for a given underlying set? In other words, what do the fibres of $C\to Set$ look like? This is definitely an interesting question for including in a project.

As Peter points out in his comment, a space is bimorphic to the indiscrete space on the same set of points. But it is also bimorphic to the discrete space on the same set of points, and these form the top and bottom respectively of the poset of topologies on a set. This poset is very interesting, and has interesting topologies. This sort of behaviour would crop up with probably any <a href="http://ncatlab.org/nlab/show/topological+concrete+category">topological concrete category</a> over $Set$ (beware that these are often called 'topological categories' in the literature, but are neither categories internal to $Top$ nor $Top$-enriched categories)