I have a differential equation
$$\frac{d^4y}{dx^4}+(ia-2\beta^2)\frac{d^2y}{dx^2}+(\beta^4-ia\beta^2)y=0$$
The characteristic equation is
$$r^4+(ia-2\beta^2)r^2+(\beta^4-ia\beta^2)=0$$
and the roots of the characteristic equation are
$$r_{1,2}=\pm\beta, r_{3,4}=\pm(\beta^2-ia)^{\frac{1}{2}}$$
I know the fundamental solutions corresponding to $r_{1,2}$ are $C_1e^{\beta z}$ and $C_2 e^{-\beta z}$. My question is what are the corresponding fundamental solutions for $r_{3,4}$? My concern or doubt is why all the books just give the general solution for the root of characteristic equation like $r=a\pm bi$ but not for $r=\pm(a \pm bi)^{\frac{1}{n}}$?
Thanks you for any advise in advance.