There is another example. Take three distinct complex numbers $z_1,z_2,z_3$ and let $X$ be the Riemann surface obtained by integrating the form $\sqrt{(z-z_1)(z-z_2)(z-z_3)}dx$. The map $g: X \to\mathbb C$ which sends each germ $\varphi_z\in X$ to $z$ is a covering with infinitely many sheets above the $z$-plane. (The ramification points are exactly above $z_1,z_2,z_3$, and there infinitely many; each one is of order 2. This is so because near $z_k$ the form is like $(z-z_k)^{1/2}dz$ and therefore the primitive is like $(z-z_k)^{3/2}$.) Now consider the holomorphic map $f: X\to\mathbb C$ which sends each germ $\varphi_z\in X$ to $\varphi_z(z)$. The branch points of $f$ are exactly the same as the branch points of $g$ and they are of order 3. (To see that resolve the equation $\zeta=(z-z_k)^{3/2}+cte$ in $z$.) Now comes the important point: For nearly all choices of $z_1,z_2,z_3$ the periods of $ydz$ on the elliptic curve defined by the equation $y^2=(z-z_1)(z-z_2)(z-z_3)$ are $\mathbb Q$-linearly independent. This implies that the set of the images of the branch points of $f$ is dense in the plane.
In this way you can construct an example for many other 1-form having periods that are $\mathbb Q$-linearly independent...