Question: How to paint 3D plots with a sophisticated colorfunc?

I've been tasked with generating "phase plots" which are visualizations of complex functions. 

A 2D phase plot is easy to create: Given a complex function F : C -> C colour points (x, y) in R^2 by [ arg(F(x+I*y)), 1, 1, colortype=HSV].   Something like the following seems to do the trick
    
    p := plot3d( 
        1,
        x=-5..5,
        y=-5..5,
        scaling=constrained,
        color=[ argument(f(x+I*y))/2/Pi, 1, 1, colortype=HSV],
        axes=none,
        style=patchnogrid,
    ):
    
    g := plottools[transform]((x,y,z)->[x,y],p);
    plots[display]( g(p) ):

Now, "colouring each point of R^2" is only possible using some type of bijection onto the Riemann sphere or Pseudosphere.

The pseudosphere is:

    x := (u,v) -> sech(u)*cos(v);
    y := sech(u)*sin(v);
    z := u - tanh(u);
    
    return  plot3d(
        [x(u,v),y(u,v),z(u,v)],
        u=0..3,
        v=0..2*Pi,
        numpoints=2^10,
        lightmodel=none,
        color=ColorFunc(u,v) );

In this case I need  ColorFunc to be:

ColorFunc := proc(u,v)
    x, y := v, exp(u);
    ans  := Re( f(x+I*y) );
    if ans > 2*Pi then
       return [0,0,0,colortype=HSV];
    end if;
    return [ans/2/Pi,1,1,colortype=HSV];
end proc;

But it seems that "ColorFunc" cannot be very sophisticated.  Namely, it cannot contain "frems" or even "if" statements because (as far as I can tell) of the order of evaluations.  

It seems possible that I can generate a psuedosphere then change colours AFTER by swapping out the COLOR information in a more sophisticated way.  How can I do this?  I really just need to know how to identify and swap out points from a MESH.