Quaternion interpolation is involved in animation. In Poser, you have 3 types of interpolation; constant, linear, and spline (or quaternion). Constant is just that; define a simple motion and it does it. Linear is also simple; set a motion from point A to point B and it occurs at a steady speed (that speed being determined by how many frames it takes to move between the two points).

Quats are more complicated; applied to the point A-point B motion, the motion begins slow at point A, increases in speed as it nears the centerpoint of the distance, and slows down to stop at point B. This effect is more realistic in terms of what human motion is. The reason it's provided as a checkable option is the incredible trouble it can get  you into. If you look at spline based interpolation on the Poser graph editor, you see that the motion curve looks like half of a sine wave....which is exactly what it is (or more accurately, half a sine wave with the bottom half clipped). Trouble comes when you decide to adjust a spline by adding a new keyframe and moving it. The new keyframe introduces a new sine wave into the motion pattern...and one that is not clipped off at the bottom. So instead of a simple, gentle curve, you can get a sudden spike of motion far above your intended distance....and and equally deep drop into negative motion numbers before it returns to the initial keyed values. If you've ever seen what looks like a smooth animation suddenly explode into chaos and incredibly shredded geometry, you most likely just watched someone mess up quaternion keyframing.

Vue can get cranky with the values that some bad quats give (you get into a lot of floating point math with it), so they made it an option. If you uncheck it,  Vue treats the animation as linear keyframed.

And one hopes the lady with the bumper tag fetish got a visit from the nice police, considering that was attempted murder and assault with a deadly weapon......