Plot3D with coloring based on phase value of f

If you use ColorFunctionScaling -> False for both plots you will get near agreement:

f[z_] := (z - 1)/(z^2 + z + 1)

GraphicsRow[{
  {Plot3D[Log[Abs[f[x + I y]]], {x, -2, 2}, {y, -2, 2}, 
    ColorFunction -> Function[{x, y, z}, Hue[Arg[f[x + I y]]]], 
    ColorFunctionScaling -> False, Mesh -> None, 
    BoxRatios -> {1, 1, 1}, 
    ViewPoint -> {0, 0, Infinity}]},
  {DensityPlot[
    Arg[f[x + I y]], {x, -2, 2}, {y, -2, 2}, ColorFunction -> Hue, 
    ColorFunctionScaling -> False]}
}]

With a manually scaled color function and more PlotPoints:

GraphicsRow[{
  {Plot3D[Log[Abs[f[x + I y]]], {x, -2, 2}, {y, -2, 2}, 
    ColorFunction -> Function[{x, y, z}, Hue[Arg[f[x + I y]]/10]], 
    ColorFunctionScaling -> False, Mesh -> None, 
    BoxRatios -> {1, 1, 1}, ViewPoint -> {0, 0, Infinity}, 
    PlotPoints -> 200]},
  {DensityPlot[
    Arg[f[x + I y]], {x, -2, 2}, {y, -2, 2}, 
    ColorFunction -> (Hue[#/10] &), ColorFunctionScaling -> False]}
}]

It does not seem to have been touched properly, so: what ColorFunctionScaling does is exactly what it says on the tin; if you have some color function like Hue[Arg[x + I y]], then any x and y fed to it is scaled beforehand such that the largest x or y is mapped to $1$, and the smallest x or y is mapped to zero. Witness the difference between these two plots:

{DensityPlot[x, {x, 0, 2}, {y, 0, 1}, AspectRatio -> Automatic, 
             ColorFunction -> Hue, ColorFunctionScaling -> True], 
 DensityPlot[x, {x, 0, 2}, {y, 0, 1}, AspectRatio -> Automatic, 
             ColorFunction -> Hue, ColorFunctionScaling -> False]} // GraphicsRow

Spot the difference? In the first plot, due to the scaling, the entire $(0,2)$ range is remapped to $(0,1)$ before being fed to the color function, which is why we only see a single band of hues. In the second plot, without the scaling, Hue[] is evaluated over the range $(0,2)$, which is why you see two bands (recall that Hue[] is $1$-periodic).

Now, to your main problem: the nutty thing is that Hue[]'s natural domain is $[0,1)$, while Arg[] gives output in the range $(-pi,pi]$. Thus, a rescaling has to be done:

Plot3D[Log[Abs[f[x + I y]]], {x, -2, 2}, {y, -2, 2}, AxesLabel -> {"x", "y", "z"}, 
       ColorFunction -> Function[{x, y, z}, Hue[Rescale[Arg[f[x + I y]], {-π, π}]]],
       ColorFunctionScaling -> False, Mesh -> None, BoxRatios -> {1, 1, 1}]

Compare now with your DensityPlot[] of the phase.

If, however, you want the convention where red corresponds to values on the real axis, then you do this:

Plot3D[Log[Abs[f[x + I y]]], {x, -2, 2}, {y, -2, 2}, AxesLabel -> {"x", "y", "z"}, 
       ColorFunction -> Function[{x, y, z}, Hue[Mod[Arg[f[x + I y]]/(2 π), 1]]],
       ColorFunctionScaling -> False, Mesh -> None, BoxRatios -> {1, 1, 1}]

Compare with the image from the book:

---

While I'm at it: I've grown to become fond of the phase coloring suggested by the DLMF (the link also explains why one might prefer to use their color scheme instead of the usual color wheel). Here's how you might apply it to your "analytic landscape":

With[{hf = Interpolation[Transpose[{Range[0, 1, 1/4], {0, 1/6, 1/2, 2/3, 1}}],
                         InterpolationOrder -> 1]},
 DLMFContinuousColorPhase[u_?NumericQ, rest___?NumericQ] := Hue[hf[Mod[u/(2 π), 1]], rest]]

Plot3D[Log[Abs[f[x + I y]]], {x, -2, 2}, {y, -2, 2}, 
       AxesLabel -> {"x", "y", "z"}, BoundaryStyle -> None, BoxRatios -> {1, 1, 1}, 
       ColorFunction -> (DLMFContinuousColorPhase[Arg[f[#1 + I #2]]] &), 
       ColorFunctionScaling -> False, Mesh -> None]