Constructing a quasiregular analogue of $z \exp(z)$ in dimension 3

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.