We know that every complex number has exactly $n$ $n$-th roots in the complex plane, and we usually take (if the context where we are working doesn't tell us more) the one with real and imaginary part both positive at the same time as the "principal root".
Now this works easily when the index of the root is small but what should I take as -for example- the principal $12$-root of $-1$ ?
There are $3$ roots in the first quarter of the complex plane so what "rule" should I follow to choose the principal branch of the $12$-root function ?
It is not true that for every complex number $z$ and every positive integer $n$ that $z$ has an $n$th root with real and imaginary part both positive. Even for $z = 1$ there is no such root for $n \leq 4$. Even if we weaken the condition to ask that both be nonnegative (perhaps this is what was intended), this is still not true: For example, the square roots of $-i$ are $\pm\left(-\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\right),$ neither of which have both real and imaginary part nonnegative. On the other hand, there are two $5$th roots of unity ($1$ and $\frac{1}{4}\left[(-1 + \sqrt{5}) + i \sqrt{10 + 2 \sqrt{5}}\right]$) with both parts nonnegative.
As for the question:
We intepret an exponential expression $z^{\alpha}$ with nonintegral exponent as the quantity $$\exp(\alpha \log z),$$ which in particular depends on a choice of branch of logarithm function. If we declare what we mean by principal branch of the logarithm function,, we determine a principal value of $z^{\alpha}$ for all $\alpha$, and in particular for the reciprocals $\alpha = \frac{1}{n}$, $n \in \Bbb Z_+$, that is for the $n$th root functions.
My convention for logarithm is that the principal branch is the one for which the argument takes values in $(-\pi, \pi]$, so that the branch cut lies along the negative real half-axis. In particular, for $\alpha \in (0, 1)$, $$z^{\alpha} = \exp(\alpha \log z) = |z|^{\alpha} e^{i \alpha \arg z}$$ has argument in $(-\alpha \pi, \alpha \pi]$. By construction, this condition (which determines a half-open sector of central angle $2 \pi \alpha$ centered along the positive real half-axis) determines a unique value $z^{\alpha}$ and in particular, for $\alpha = \frac{1}{n}$, a unique "principal" $n$th root of any nonzero complex number.