Start with some 2D geometry and complex number examples.
Let the point {$p=(1,2)$}. The corresponding complex number is {$c=1+2i$}. Suppose you want to rotate it by {$\theta=90^\circ=\pi/2$} radians. That is equivalent to multiplying {$c$} by {$e^{i\theta}=e^{i\pi/2}=i$}.
So, {$c'=c e^{i\theta} = (1+2i)i = -2+i$}.
The corresponding 2D point is {$(-2,1)$}.
Now to quaternions in general.
Let {$q_1=(1,2,0,0)$} and {$q_2=(3,0,4,0$}. {$q_1+q_2=(4,0,4,0)$}.
{$q_1 q_2 = xxxxxx$} {$q_2 q_1 = xxxxxx$}, which is different.
Now to 3D and quaternions.
For example, the 3D point {$(1,0,2)$} corresponds to the quaternion {$1i+0j+2k$}.
to be continued.