The two quaternions $\omega={1\over 2}(-1,1,1,1)$ and $q={1\over 4}(0,2,\sqrt{5}+1,\sqrt{5}-1)$ generate a finite group under multiplication with 120 elements that form the vertices of a 600 cell, when considered as vectors of ${\mathbb R}^4$.
I tried to find a similar pair of quaternions that generate the dual polytope, the 120 cell. Since it is not mentioned on the wiki-page on quaternions and I couldn't find it somewhere else, I assume that it is not possible to find such a pair of quaternionic generators?
Is there an easy explanation, why it works for the 600 cell and not for the 120 cell?
Summarizing the discussion in the comments.
The binary icosahedral group $2I$ in the quaternions form a 600-cell with 120 vertices. The dual polychoron, the 120-cell with 600 vertices, is not a group, so it doesn't make sense to talk about generators.
What is true, though, is that there is a free action of $2I$ on it by left- (or right- or two-sided) multiplication which is free with five orbits, realizing the 120-cell as a compound of 5 inscribed 600-cells (see my question about this).