@LichengZhang The English word "next" as used by John May implies something in the future. Maple 2019.2's release is something in the past.

Are you saying that after 6 years on MaplePrimes, you don't know how to even enter the above equations into Maple?

@Kitonum Please post worksheets inline so that I can read them on my phone. 

Please post worksheets inline so that I can read them on my phone.

@dingtianlidi Hmm, I see that you're right. I couldn't do anything to get colorscheme= ["xyzcoloring", ...] to work with tubeplot. But it won't give an error message either! Unfortunately, that misled me to believe that my previous plot was correct.

Here's an alternative which I'm sure works (as can be seen in the plot below). We use the old-style option color= COLOR(HUE, C(t)) where C(t) is an expression in the parameter t scaled to 0..1.
 

Download ColoredTubeplot.mw

Note that x+y+z is constant 5 on the circle, so that doesn't make a good coloring function.

The radius option to tubeplot refers to the cross-sectional radius of the tube, not the radius of the overall circle.

@Carl Love Even without the wireframe, the use of cylindrical coordinates for the surface eliminates the gap between the upper and lower surfaces, even at modest resolution (I used 53x53 above).

There must be an underlying implementation of Maple supporting this Mobius. To what extent does the Mobius user have access to that? It would be easy to write a quaternion package for Maple. Would you be able to install and access it?

@zphaze I formally reported the bug.

It's not a new feature. Loops have always been that way. 

@ActiveUser All parts of Maple"s looping command that come before the word do are optional and always have been. 

@Scot Gould While it is true that the originally posted worksheet is missing a division operator in the parameters section of its Explore command, this is not the cause of the errors shown,. If you compare your worksheet with the OP's, you'll see that the latter has no boundary markers separating the regions of expository text from the executable code. (These markers can be seen in the extreme left margin of almost any worksheet.) I don't know whether this is the OP's doing, or whether the worksheet got mangled by MapleCloud.

And this is the same plot with the only difference being that I turned off transparency:

Here is the plot done in cylindrical coordinates. By using this, it's possible to get just the radial lines of the grid, plus a border. In my opinion, the syntax makes it much easier to color a grid (aka wireframe) in a plot3d than it is to color a spacecurve. In the plot below, the radial lines are colored by the imaginary part.

restart
:
f:= (r,theta)-> r*cos(theta/2): #Re(sqrt(z))  
Square:= piecewise(
    theta < -3*Pi/4, -sec(theta), #x = -1
    theta < -Pi/4,   -csc(theta), #y = -1
    theta < Pi/4,     sec(theta), #x =  1
    theta < 3*Pi/4,   csc(theta), #y =  1
                     -sec(theta)  #x = -1
):
plots:-display(
    plot3d(
        (common:= 
           [seq([r, theta, k*f(r,theta)], k= [-1,1])], 
           r= 0..Square, theta= -Pi..Pi, coords= cylindrical
        ), 
        grid= [53$2], style= surface, glossiness= 1, 
        transparency= .002,    #close to the min value that does anything
        colorscheme= [         #adaptation of your original coloring:
            "xyzcoloring", 
            [(x,y,z)-> y^2, (x,y,z)-> abs(x*y), (x,y,z)-> x^2]
        ]
    ), 
    plot3d(
        common,
        grid= [2,23],                     #Shows only the radial lines. 
        style= wireframe, thickness= 3, 
        color= abs(r*sin(theta/2))             #abs(Im(sqrt(z)))  
    ),
    lightmodel= light2, orientation= [55,75,0], projection= .85, axes= frame,
    labels= ['x', 'y', Re(w)]
);

@Stretto There are 31 predefined 3d coordinate systems, and you can also define your own. These are documented on help page ?coords. Of course, the most important are rectangular (default), cylindrical, and spherical. You can use them in a 3d plot by using the coords option. I'll post an example (your Riemann surface in cylindrical coordinates) as a followup to my Answer.

Note that Acer used cylindrical coordinates by manually doing the appropriate coordinate transformation and plotting in default rectangular coordinates. But by explicitly using coords= cylindrical, it's possible to get just the radial lines plus a border.

@Kitonum Maple 12 doesn't have the key option to sort.