Minimum number of real multiplications to multiply two quaternions
de Groote showed that one could compute both $uv$ and $vu$ using only a total of 10 non-scalar multiplications.

Howell and Lafon, in a tech report (that I could not find a refereed version of and have not checked myself, but is probably correct) showed that one can do quaternion product with only 8 multiplications. 

7 is certainly a lower bound, as noted in both of the above papers.

(While this was an interesting question and fun to look up, I found the answer in about 5 minutes of Googling. Google is your friend.)