  We construct a quasiregular analogue of the function $z\exp(z)$ in dimension
3, which gives the first explicit example of a quasiregular mapping of
transcendental type that has exactly one zero. We then modify the construction
to create a family of such quasiregular mappings and study their dynamics. From
this, we also construct the first quasimeromorphic mappings with an essential
singularity at infinity where the backward orbit of infinity is non-empty and
finite.
