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? $
On
Since @LordSharktheUnknown discussed the trigonometry, I'll answer your later questions. Do you want $w:=z_1/z_2$ to satisfy $z_1=z_2w$ or $z_1=wz_2$? It matters, which is why we don't usually write such expressions as $\frac{i+j}{k}$; you'd want to say $(i+j)k^{-1}$ or $k^{-1}(i+j)$ instead. (These are respectively $-ik-jk=ki-jk=j-i,\,i-j$.) That quaternions don't commute also introduces problems with defining exponentiation. Do we want $z_1^{z_2}$ to mean $\exp(z_2\ln z_1)$ or $\exp((\ln z_1)z_2)$?
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 $bi+cj+dk$.