Argand formula and more for quaternions?

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 $bi+cj+dk$.

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 $\exp((\ln z_1)z_2)$?