Yes, you are mistaken. You need 3 dimensions to represent a 3D rotation (2 for picking an axis, plus one for picking an angle). To represent translations and rotations together, you need 6 dimensions.
This also shows that the talk of needing to visualize 4d to understand quaternions is disingenuous. The formula for using a quaternion to rotate a vector is qvq^-1, from which it is immediate that changing the length of a quaternion does not change the rotation it represents. So you can just deal with unit-length quaternions, which form a 3D space.
Hmm, I seem to be mixing up using 4d linear operations (ie, a 4x4 matrix) to represent translation, rotation, scaling, etc, with their combinations, with representing a rotation itself. Thanks for your help—I'll dig more into this.
This also shows that the talk of needing to visualize 4d to understand quaternions is disingenuous. The formula for using a quaternion to rotate a vector is qvq^-1, from which it is immediate that changing the length of a quaternion does not change the rotation it represents. So you can just deal with unit-length quaternions, which form a 3D space.