Argand formula and more for quaternions? Is it possible to define a similar form of Argand's formula but for quaternions?
In the sense 
$$ \cos(nA)+i\cos(nB)+j\cos(nC)+k\cos(n) =(\cos(A)+i\cos(B)+j\cos(C)+k\cos(D))^{n}, $$
where $A, B, C, D$ are the angles of the quaternion with respect the axes $x,y,z,t.$
Also for a quaternion why can you not define the 'numbers'
$ \frac{i+j}{k} $ or $ i^{j+k} $ ... where the quaternion is defined in 4-dimensions as
$ a+ib+cj+dk =z? $

A unit quaternion can be written as
$$q=\cos t+(bi+cj+dk)\sin t$$
where $b^2+c^2+d^2=1$. Then
$$q^n=\cos nt+(bi+cj+dk)\sin nt.$$
This just follows from the usual complex case: there's an isomorphism between
$\Bbb C$ and the subalgebra generated by $bi+cj+dk$, taking $i$ to
$bi+cj+dk$.
