  In this article, we study the boundedness and several properties of the
quaternion Wigner transform. Using the quaternion Wigner transform as a tool,
we define the quaternion Weyl transform (QWT) and prove that the QWT is compact
for a certain class of symbols in $L^{r}\left(\mathbb{R}^{4},
\mathbb{Q}\right)$ with $1 \leq r \leq 2.$ Moreover, it can not be extended as
a bounded operator for symbols in $L^{r}\left(\mathbb{R}^{4},\mathbb{Q}\right)$
for $2<r<\infty.$ In addition, we prove a rank analogue of the
Benedicks-Amrein-Berthier theorem for the QWT. Further, we remark about the set
of injectivity and Helgason's support theorem for the quaternion twisted
spherical means.
