How can I recreate Trott's Riemann Surface plot in Mathematica?

Natively in version 9, you can do the following:

ParametricPlot3D[{r Cos[phi], r Sin[phi], Sqrt[r] Sin[phi/2]}, {r, 0, 1}, 
 {phi, 0, 4 Pi}, PlotPoints -> {20, 60}, Boxed -> False, Axes -> False,
  ColorFunction -> (Hue[#5/(4 Pi)] &), ColorFunctionScaling -> False]

Alternatively, you can always use the exact code using the V5 emulator:

<<Version5`Graphics`
ParametricPlot3D[{r Cos[phi], r Sin[phi], Sqrt[r] Sin[phi/2], 
  SurfaceColor[Hue[phi/(4 Pi)]]}, {r, 0, 1}, {phi, 0, 4 Pi}, 
  PlotPoints -> {20, 60}, Boxed -> False, Axes -> False]

Of course, the graphics aren't quite as nice. You can go back to the newer style graphics as follows:

<<Version6`Graphics`

As of Version 6, SurfaceColor has been superseded by Specularity and Glow.

One could specify the variables explicitly like here :

ParametricPlot3D[{ r Cos[φ], r Sin[φ], Sqrt[r] Sin[φ/2]}, {r, 0, 1}, {φ, 0, 4 π},
   ColorFunction -> Function[{x, y, z, r, φ, θ},
                      {Specularity[#], Glow[#]}& @ Hue[Rescale[φ, {0, 1}]]],
   PlotPoints -> {20, 60}, Boxed -> False, Axes -> False]

or adding a more thrilling variation of ColorFunction (singularity when r -> 0) :

ParametricPlot3D[{ r Cos[φ], r Sin[φ], Sqrt[r] Sin[φ/2]}, {r, 0, 1}, {φ, 0, 4 π},
    ColorFunction -> Function[{x, y, z, r, φ, θ},
                       {Specularity[#], Glow[#]}& @ Hue[ Rescale[ φ/r, {0, 1}]]],
    PlotPoints -> {20, 60}, Boxed -> False, Axes -> False]

...and now, for something that takes a bit of the old, and a bit of the new:

ColoredMakePolygons[vl_List] := Module[{msh = Map[Most, vl, {2}], cols, dims},
        cols = Map[First[Cases[Last[#], _?ColorQ, {0, Infinity}]] &, vl, {2}];
        cols = Map[If[Head[#] === GrayLevel, #, ColorConvert[#, RGBColor]] &,
                   cols, {2}];
        dims = Most[Dimensions[msh]];
        GraphicsComplex[Apply[Join, msh], Polygon[Flatten[Apply[
                        Join[Reverse[#1], #2] &, 
                        Partition[Partition[Range[Times @@ dims], Last[dims]],
                                  {2, 2}, {1, 1}], {2}], 1]],
                        VertexColors -> Apply[Join, cols]]] /;
ArrayDepth[vl] == 3 && Last[Dimensions[vl]] == 4

With[{m = 21, n = 61}, 
     Graphics3D[ColoredMakePolygons[
                N @ Table[{r Cos[φ], r Sin[φ], Sqrt[r] Sin[φ/2], Hue[φ/(4 π)]},
                          {r, 0, 1, 1/(m - 1)}, {φ, 0, 4 π, 4 π/(n - 1)}]], 
                Boxed -> False, Lighting -> "Neutral"]]

Reference link:

   Manipulate[
     ParametricPlot3D[
      Evaluate@{Re[(1 - α) (r Exp[
               I φ])^2 + α (r Exp[I φ])^3], 
        Im[(1 - α) (r Exp[I φ])^2 + α (r Exp[
               I φ])^3], r Cos[φ]}, {r, 0, 
       2}, {φ, -Pi, Pi}, PlotRange -> All, Mesh -> 20, 
      ColorFunction -> (Hue[#5] &), PlotPoints -> 100, MaxRecursion -> 1, 
      BoxRatios -> {1, 1, 1}, PlotRange -> All, Axes -> False, 
      ImageSize -> {435, 435}, Boxed -> False], {{α, 0, "%"}, 0, 
      1}]

https://en.wikipedia.org/wiki/Riemann_surface