A Cayley graph is a $1$-dimensional cell complex associated to any presentation of a group $G$, where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (connecting two elements). A Cayley complex is a $2$-dimensional generalization, where the $2$-cells are given by the relations (here is a nice YouTube video on it by Daniel Tubbenhauer).

Question: What about a generalization of the Cayley complex to higher dimensions?

I am looking for obstructions preventing such a generalization to all the finitely generated (or finitely presented) groups, and/or references discussing such a generalization to all or a large class of them.

Here is my effort to see what such a generalization might look like.