Comparing the results in parts (1) and (2) we see that in part (1)

q = \cos 45^{o} + \sin 45^{o} i

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 $q = \cos θ + \sin θ k$ the map $q v q^{- 1}$ fixes the z-axis and gives a rotation of $2 θ$ about the z-axis.

For a more general account of quaternions and rotations see the article....