No such tangent vector field exists for the ordinary 2-sphere. Clearly the 1-sphere has such a vector field. If Q thus induces rotation R, then Q1 Q2 induces rotation R1 R2. If a rotation R carries U into U', then a quaternion corresponding to R carries U1 into U1'. This shows that in the quaternion representation of the rotation group, the Haar measure is the Lebesgue measure on the 3-sphere.
Every rotation is a rotation by some angle theta about some axis.
2 pi Quaternions form a convenient 4-parameter representation of rotations, since composition of rotations is done by quaternion multiplication. In space guidance systems under development at D-lab, the attitude of the spacecraft is stored in the guidance computer as a quaternion.