Without benchmarks, I do not trust you. :) have you tried? On what platform? And did you take a look at the assembly code?
Its geometric product between motors is the equivalent of the 4x4 matrix product.
Although the basic computational complexity tells the story :
4x4 matrix product : 16 floats storage, 64 multiplies, 48 additions.
3D PGA versor product : 8 floats storage, 48 multiplies, 40 additions.
> If the quaternion is used to rotate several points (>1) then it is much more efficient to first convert it to a 3x3 Matrix. Comparison of the operation cost for n transformations:
> - Quaternion2: 30n
> - Via a Matrix3: 24 + 15n
So really it depends on what you're doing (admittedly here it is 3x3 not 4x4)
[0] https://eigen.tuxfamily.org/dox/classEigen_1_1QuaternionBase...