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.