Status quo of category theory and monads in theoretical computer science research?
There have been a number of developments with regards to the use of monads in the theory of computation since Eugenio Moggi's work. I am not able to give a comprehensive account, but here are some points that I am familiar with, others can chime in with their answers.

Specific examples of monads

You do not have to study super-general theory all the time. There are examples of monads that are very interesting and sufficiently complicated to fill an entire undergraduate thesis.

I like very much Dan Piponi's blog where he gives amazing examples of how monads can be used to mix functional programming and mathematics. Search for his work on knots and braid through monads, for example.

Another specific example of mondas worth studying was given by Martin Escardo and Paulo Oliva in the context of selection functionals, see their Selection Functions, Bar Recursion, and Backward Induction, or perhaps to get interested first read What Sequential Games, the Tychonoff Theorem and the Double-Negation Shift have in Common (associated Haskell and Agda files here).

Mathematical background

Monads come from category theory and are much older than Eugenio Moggi. You could study the background theory if you are mathematically inclined. For instance, you could attack Beck's monadicity theorem. A theoretical computer scientist can never know too much math.

Variations on a theme

You could look at something that is not strictly monads.

For instance, Connor McBride and Ross Paterson's Idioms: applicative programming with effects shows how one can generalize monads to something that is practically relevant and insightful.

Or you could look at how comonads are used to model computational effects. Someone should suggest some references for this topic, but a good start might be David Overtone's slides.

Modal type theory

In homotopy type theory, as well as in type theory in general, monads appear in the shape of modal type theory. Recently modal type theory has been considered in homotopy type theory because the truncation operators are examples of modal operators. And then there is cohesive homotopy type theory in which modal operators (which are monads) play an essential role.

Algebraic effects and handlers

[Disclaimer: partially blowing my own horn here.]

A while ago Gordon Plotkin and John Power obserrved that many computational effects are not just any monads, but special monads arising from algebraic theories. This lead to a whole new treatment of computational effects known as algebraic effects. Later Gordon Plotkin and Matija Pretnar introduced handlers and together with algebraic effects they form a very nice theory of computational effects. One advantage of this approach is that algebraic theories can be easily combined while monads cannot.

You could study how exactly algebraic effects relate to monads. You could look at how people implemented algebraic effects and handlers, say in the Eff language or in Haskell as a library. This is more or less current research.