According to the wikipedia page on Quaternions:
The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices.
However, I have to admit, I don't fully understand what quaternions are and why they are useful. I have tried to read the article, but I don't understand why defining such a system is useful.
It appears to a define a four dimensional space in which 3 components are imaginary and one is real. Is this attempting to describe spacetime?
Regardless, I was hoping someone here could show how to represent a rotation using both quaternions and matrices and compare the two for me.