Does this complex root always exist?

Let $z=r_1e^{i\theta}$, $w=r_2e^{i\gamma}$ where $r_1,r_2,\theta,\gamma$ are real. Comparing the moduli of $z^n, w^m$, we see that $r_1^n=r_2^m$, so as you notice there exists real $r$ such that $r_1=r^m, r_2=r^n$.

Comparing the arguments, we see that $n\theta-m\gamma=2\pi k$ for some integer $k$.

Since gcd$(m,n)=1$, we have $mv+nu=1$ for some integers $v,u$.

Then we can take $t=re^{i(\theta/m-2\pi ku/m)}$. Indeed $t^m=z$. Also notice that $$\theta/m-2\pi ku/m=\gamma/n+2\pi k/(mn)-2\pi ku/m=\gamma/n \\ +2\pi k (1-un)/mn=\gamma/n+2\pi k(mv/(mn))=\gamma/n +2\pi k v/n.$$ So $t^n=w$.