I recently took the plunge and learned unit quaternions and dual quaternions for representing rotations, compound rotation-translations, and kinematic chains and geometric interpolation (serial and parallel robot arms, character skeletons, skinning).

The advantages I found are:

Unit quaternions represent rotations with less redundancy than matrices

It's easier and more intuitive to derive, manipulate, simplify, interpolate, and solve quaternions than matrices.

They're less abstract/more concrete than matrices.

Rotating with quaternions takes fewer multiplications than matrices.

Dual quaternion inverse kinematics are easier to derive and faster than matrices

Unit and Dual quaternions have an efficient implicit form, which further speeds up IK. See: https://www.researchgate.net/profile/Neil-Dantam/publication...