I am trying to understand if the hypergeometric function with the following choice of parameters has any branch cut in the complex plane $$ \ _{2}F_{1}\left(a,b - i \rho ,c ,1-z^{2}\right) $$
here $a,b,c \in \mathbb{N}$ and $\rho \in \mathbb{R}$ and $\rho > 0$.
I did some numerical explorations with Mathematica and I was not able to find any branch cuts on the real line. Is this property true?
I was not able to find any reference on the branch cuts of the hypergeometric function with complex parameters.
Complex parameters have nothing to do with the choice of branch cuts. The conventional way to fix the branch cut on the principal sheet of the Riemann surface of $_2F_1(a,b;c;x)$ is to draw it from $1$ to $+\infty$ along the real axis in the complex plane of $x$. For example, this is how $_2F_1$ is implemented in Mathematica.
In terms of your $z$-variable, this branch cut corresponds to two halves of the imaginary axis. Specifically, $z\mapsto 1-z^2$ maps each of the half planes $\Re z\gtrless0$ to $\mathbb C\backslash[1,+\infty)$. So the signature of the branch cut will be the difference between the values of $_2F_1(1-z^2)$, calculated for $z=\pm i\alpha+0$ (or $z=i\alpha\pm 0$) for $\alpha\in\mathbb R$.