Visualizing a normalized hyper-elliptic curve as a honey-flow on a multi-doughnut

Trying to visualize in 3D the normalized model $$y^2=f(x)$$ of a hyper-elliptic curve AND the degree two ramified cover $$(X,Y)\mapsto X$$, $$\mathbb{C}\times \mathbb{C}\to \mathbb{C}$$. Is this projection amenable to viewing the RS with the $$g$$ holes stacked, like a multi-doughnut with a honey flow on it (like in an article by Witten; forgot which), with the above projection x factoring through $$(X=x+iy,Y)\to (x+iy, |Y|)$$, to bring it in $$\mathbb{R}^3$$? I would like to see the ramification points (zeros of $$f$$) as in a Morse Theory analysis, related to the above standard projection of its normal model; something like in the sketch below: z-axis Flow on a normal model of a RS