I've noticed that rotation can be interpreted in 2 different manors. Does rotation occur about an axis, or through a pair of axes?
What I would like to know is in 4D are there 4 fundamental rotations (one about each axis) or are there 6 fundamental rotations (one about each pair of axes)?
If one of these interpretations is rotation, what is the name for the other?
A rotation always acts on a plane. So there are 6 different rotations in $\mathbb{R}^4$ that keep two axis fixed and rotate the plane spanned by the other two. You can't rotate around an axis unless you are in $\mathbb{R}^3$. To see this consider the plane $\mathbb{R}^2$, there is only one rotation of the entire plane, not two for the two axis.