  The paper explores further the computation of the quaternionic numerical
range of a complex matrix. We prove a modified version of a conjecture by So
and Tompson. Specifically, we show that the shape of the quaternionic numerical
range for a complex matrix depends on the complex numerical range and two real
values. We establish under which conditions the bild of a complex matrix
coincides with its complex numerical range and when the quaternionic numerical
range is convex.
