Taking as departure point an article by Cameron, Gadouleau, Mitchell and Peresse on maximal lengths of subsemigroup chains, we introduce the subsemigroup complex H(S) of a finite semigroup S as a (boolean representable) simplicial complex defined through chains in the lattice of subsemigroups of S. The rank of H(S) is the above maximal length minus one and H(S) provides other useful invariants concerning the lattice of subsemigroups of S. We present a research program for such complexes, illustrated through the particular case of combinatorial Brandt semigroups. The results include alternative characterizations of independence and bases, asymptotical estimates on the number of bases, or establishing when the complex is pure or a matroid. This is joint work with Stuart Margolis (Bar-Ilan University, Ramat Gan, Israel) and John Rhodes (University of California, Berkeley, USA).