  Two words are $k$-binomially equivalent whenever they share the same
subwords, i.e., subsequences, of length at most $k$ with the same
multiplicities. This is a refinement of both abelian equivalence and the Simon
congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps
the integer $n$ to the number of classes in the quotient, by this $k$-binomial
equivalence relation, of the set of factors of length $n$ occurring in
$\mathbf{x}$. This complexity measure has not been investigated very much. In
this paper, we characterize the $k$-binomial complexity of the Thue--Morse
word. The result is striking, compared to more familiar complexity functions.
Although the Thue--Morse word is aperiodic, its $k$-binomial complexity
eventually takes only two values. In this paper, we first obtain general
results about the number of occurrences of subwords appearing in iterates of
the form $\Psi^\ell(w)$ for an arbitrary morphism $\Psi$. We also thoroughly
describe the factors of the Thue--Morse word by introducing a relevant new
equivalence relation.
