Comparing the results in parts (1) and (2) we see that in part (1)
q = cos45o + sin45oi and the map qvq-1 fixes the x-axis and
gives a rotation of twice 45 degrees about the x-axis.
The result in (2) is slightly more general showing that where
q = cosq+ sinqk
the map q v q-1 fixes the z-axis and gives a rotation of 2q
about the z-axis.
For a more general account of quaternions and rotations see the article....