  We consider the problem of immersing the complete digraph on t vertices in a
simple digraph. For Eulerian digraphs, we show that such an immersion always
exists whenever minimum degree is at least t(t-1), and for t at most 4 minimum
degree at least t-1 suffices. On the other hand, we show that there exist
non-Eulerian digraphs with all vertices of arbitrarily high in- and outdegree
which do not contain an immersion of the complete digraph on 3 vertices. As a
side result, we obtain a construction of digraphs with large outdegree in which
all cycles have odd length, simplifying a former construction of such graphs by
Thomassen.
