ContourPlot[Im[Sqrt[(Tanh[x + I*y] - Tanh[2 x + I*2 y])^2 + (Tanh[x + I*y]
         Tanh[2 x + I*2 y] + 1)^2-1 - 2 ((Tanh[x + I*2 y])^2)((Tanh[x + I*y])^2) ]],
           {x, -10, 10}, {y, -10, 10}, AxesLabel -> Automatic,ContourShading -> Automatic,
                         ColorFunction -> "Rainbow",  Contours -> 20]

ContourPlot[Re[Sqrt[(Tanh[x + I*y] - Tanh[2 x + I*2 y])^2 + (Tanh[x + I*y]Tanh[2 x + I*2 y] + 1)^2 - 1 - 2 ((Tanh[x + I*2 y])^2) ((Tanh[x + I*y])^2) ]],
                     {x, -10, 10}, {y, -10, 10}, AxesLabel -> Automatic,
ContourShading -> Automatic, ColorFunction -> "Rainbow",  Contours -> 20]

expr = Sqrt[(Tanh[z]-Tanh[2z])^2+(Tanh[z] Tanh[2z]+1)^2-1-2 Tanh[z]^2Tanh[2z]^2];

pts = ComplexAnalysis`BranchPoints[expr, z]

Simplify[pts, C[1] ∈ Integers]

ComplexAnalysis`BranchCuts[expr, z]

With[{z = x + I y},
  expr = (Tanh[z] - Tanh[2 z])^2 + (Tanh[z] Tanh[2 z] + 1)^2 - 1 - 2 ((Tanh[2 z])^2) ((Tanh[z])^2);
  branchCutRegion[x_, y_, __] = Re[expr] <= 0;
];

bpvals = Union[{x, y} /. Solve[(expr == 0 || 1/Together[TrigToExp[expr]] == 0) && -10 < x < 10 && -10 < y < 10, {x, y}]];

ContourPlot[Im[expr] == 0, {x, -10, 10}, {y, -10, 10},
  RegionFunction -> branchCutRegion, PlotPoints -> 100,
  Epilog -> {Red, Point[bpvals]}
]

binnedabs = Compile[{{z, _Complex}},
  Module[{f, abs, rnd, sgn, val},
    f = (Tanh[z] - Tanh[2 z])^2 + (Tanh[z] Tanh[2 z] + 1)^2 - 1 - 2 Tanh[2 z]^2 Tanh[z]^2;
    abs = Abs[f];
    rnd = Round[abs, .2];
    val = If[rnd == 0, f, rnd Sign[f]];
    {
      Divide[Mod[Arg[val], 2π], 2π],
      Power[1 + 0.3*Log[Abs[val] + 1], -1],
      Power[1 + 0.5*Log[Abs[val] + 1], -1]
    }
  ],
  CompilationTarget -> "C",
  Parallelization -> True,
  RuntimeAttributes -> {Listable},
  RuntimeOptions -> "Speed"
];

lattice = Array[List, {2048, 2048}, {{-10., 10.}, {-10., 10.}}].{I, 1};

raster = Raster[binnedabs[lattice], {{-10, -10}, {10, 10}}, ColorFunction -> Hue];

cutplot = ContourPlot[Im[expr] == 0, {x, -10, 10}, {y, -10, 10},
  RegionFunction -> branchCutRegion, PlotPoints -> 100, ContourStyle -> Black];

Show[
  cutplot,
  ImageSize -> 800,
  Prolog -> raster,
  Epilog -> {EdgeForm[Black], GrayLevel[.8], Disk[#, Scaled[.0045]] & /@ bpvals}
]

exprz = (Tanh[z] - Tanh[2 z])^2 + (Tanh[z] Tanh[2 z] + 1)^2 - 1 - 2 ((Tanh[2 z])^2) ((Tanh[z])^2);
exprxy = exprz /. z -> x + I y;
branchCutRegion[x_, y_, __] = Re[exprxy] <= 0;

bpvals = Union[{x, y} /. Solve[(expr == 0 || 1/Together[TrigToExp[expr]] == 0) && -10 < x < 10 && -10 < y < 10, {x, y}]];

domaincoloring = ComplexPlot[exprz, {z, -10 - 10 I, 10 + 10 I},
  ColorFunction -> "CyclicLogAbsArg", ImageSize -> 800];

cutplot = ContourPlot[Im[exprxy] == 0, {x, -10, 10}, {y, -10, 10},
  RegionFunction -> branchCutRegion, PlotPoints -> 100, ContourStyle -> Black];

Show[
  domaincoloring,
  cutplot,
  Epilog -> {EdgeForm[Black], GrayLevel[.8], Disk[#, Scaled[.0045]] & /@ bpvals}
]

domaincoloring = ComplexPlot[exprz, {z, -10 - 10 I, 10 + 10 I},
  ColorFunction -> {Hue[Divide[Mod[#8, 2π], 2π],
    Power[1 + 0.3*Log[#7 + 1], -1],
    Power[1 + 0.5*Log[#7 + 1], -1]] &, None},
  ColorFunctionScaling -> False,
  Exclusions -> None,
  ImageSize -> 800
];

myexp = Together[
  TrigToExp[
   FullSimplify[(Tanh[z] - Tanh[2 z])^2 + (Tanh[z] Tanh[2 z] + 1)^2 -
     1 - 2 Tanh[z]^2 Tanh[2 z]^2]
   ]]

Expand[Numerator[
      Together[TrigToExp[
        FullSimplify[(Tanh[z] - Tanh[2 z])^2 + (Tanh[z] Tanh[2 z] +
             1)^2 - 1 - 2 Tanh[z]^2 Tanh[2 z]^2]
        ]]]]
    mySol = z /.
       Solve[1 + 2 z^2 + 3 z^4 - 12 z^6 + 3 z^8 + 2 z^10 + z^12 == 0, z];

p1 = Show[
   Graphics[{Red,
     Point @@ {{Re[#], Im[#]} & /@ (N[Log[#]] & /@ mySol)}}],
   Axes -> True, PlotRange -> 5];
p2 = Show[
   Graphics[{Blue,
     Point @@ {{Re[#], Im[#]} & /@ (N[(Log[#] + 2 [Pi] I)] & /@
          mySol)}}], Axes -> True, PlotRange -> 15];
p3 = Show[
   Graphics[{Green // Darker,
     Point @@ {{Re[#], Im[#]} & /@ (N[(Log[#] - 2 [Pi] I)] & /@
          mySol)}}], Axes -> True, PlotRange -> 15];
Show[{p1, p2, p3}, PlotRange -> 15]