  Using the definition of colouring of $2$-edge-coloured graphs derived from
$2$-edge-coloured graph homomorphism, we extend the definition of chromatic
polynomial to $2$-edge-coloured graphs. We find closed forms for the first
three coefficients of this polynomial that generalize the known results for the
chromatic polynomial of a graph. We classify those $2$-edge-coloured graphs
that have a chromatic polynomial equal to the chromatic polynomial of the
underlying graph, when every vertex is incident to edges of both colours.
Finally, we examine the behaviour of the roots of this polynomial, highlighting
behaviours not seen in chromatic polynomials of graphs.
