The space of rotors and rotation quaternions are both three dimensional because the coordinates are restricted to unit magnitude.
Quaternions and rotors are exactly the same in practice, but the intuitions are very different. The intuition behind rotors involves planes and lines in 3D, whereas the intuition behind quaternions typically involves a unit hypersphere; there's a 3B1B video on quaternions where you can learn more about them.
You're right that the space of rotations is 3-dimensional. When I said 4, I meant without restricting to the unit hypersphere. This causes no problem for rotations because replacing q by a multiple tq leaves the formula x --> qxq^(-1) unchanged --although of course q^(-1) invloves dividing by the square of the norm so it's nice if the norm is 1.
Note that even if you do restrict q to have unit norm, q and -q still denote the same rotation! (In other words, the space of rotations isn't really S^3, but projective space RP^3.)
I'll be sure to check out the 3B1B video, I keep seeing them recommended and I think they might a good resource to recommend to my students.
Quaternions and rotors are exactly the same in practice, but the intuitions are very different. The intuition behind rotors involves planes and lines in 3D, whereas the intuition behind quaternions typically involves a unit hypersphere; there's a 3B1B video on quaternions where you can learn more about them.