  An $(m,n)$-mixed graph generalizes the notions of oriented graphs and
edge-coloured graphs to a graph object with $m$ arc types and $n$ edge types. A
simple colouring of such a graph is a non-trivial homomorphism to a reflexive
target. We find that simple chromatic number of complete $(m,n)$-mixed graphs
can be found in polynomial time. For planar graphs and $k$-trees ($k \geq 3$)
we find that allowing the target to be reflexive does not lower the chromatic
number of the respective family of $(m,n)$-mixed graphs. This implies that the
search for universal targets for such families may be restricted to simple
cliques.
