It is possible to do the symbolic multiplication $qq^*$ of a quaternion $q=a+bi+cj+dk$ by its conjugate $q^*=a-bi-cj-dk$ using Mathematica? It seems that Quaternion package only works with numeric entries.
Needs["Quaternions`"] q = Quaternion[a, b, c, d]; q ** Conjugate[q]
Quaternion[a^2 + b^2 + c^2 + d^2, 0, 0, 0]
The following links might be helpful to you: