Mobius[z_]:=I ((1+z)/(1-z));
ReIm:={Re[#],Im[#]}&

ParametricPlot[ReIm[Mobius[r Exp[I θ]]],{r,0,1},{θ,0,2π}]

ComPolar[{z_,f_}]:={Re[z],Im[z],f}

ParametricPlot3D[Table[ComPolar[{r Exp[I θ],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}]

ParametricPlot3D[Table[ComPolar[{Mobius[r Exp[I θ]],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}]

SterProj[{{x_,y_},{z_,w_}}]:={(x/(1-w)),(y/(1-w)),(z/(1-w))}
vertRot=RotationTransform[π/2,{1,0,0}]
radRot=RotationTransform[π/2,{0,0,1}]
Manipulate[
Show[ParametricPlot3D[Table[ComPolar[{r Exp[I θ],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}],
ParametricPlot3D[vertRot@radRot@SterProj[ReIm/@{r Exp[I θ], Exp[I ϕ]}],{r,0,1},{θ,0,2π}]],{ϕ,0,2π}]

ParametricPlot3D[Table[ComPolar[{Mobius[r Exp[I θ]],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}]

SterProj[{{x_,y_},{z_,w_}}]:={(x/(1-w)),(y/(1-w)),(z/(1-w))}
vertRot=RotationTransform[π/2,{1,0,0}];
radRot=RotationTransform[π/2,{0,0,1}];

Show[ParametricPlot3D[Table[ComPolar[{r Exp[I θ],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}],
ParametricPlot3D[Table[radRot@vertRot@ComPolar[{r Exp[I θ],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}]]

Torus[{a_,b_,ϕ_}]:={(1+a)Sin[ϕ],(1+a)Cos[ϕ],b}
Torus[ComPolar[{r Exp[I θ],2π n/4}]]

Show[ParametricPlot3D[Table[Torus@ComPolar[{r Exp[I θ],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}],
ParametricPlot3D[Table[radRot@vertRot@ComPolar[{r Exp[I θ],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}]]

Rot[Ω_]:=RotationTransform[Ω,{0,0,1}]

f=Table[Torus@Rot[-π/2]@ComPolar[{Mobius@(r Exp[I θ]),2π n/4}]+{0,0,π},{n,0,4}]//ComplexExpand;

ParametricPlot3D[f,{r,0,1},{θ,0,2π},PlotRange->{{-10,10},{-10,10},{-10,10}}]

Show[
ParametricPlot3D[f,{r,0,1},{θ,0,2π}],
ParametricPlot3D[Table[ComPolar[{r Exp[I θ],2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π},PlotRange->{{-3,3},{-3,3},{-3,3}}]
]

ParametricPlot3D[Table[ComPolar[{r Exp[I θ],r Sin[θ]+2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}]

f=Table[Torus@Rot[-π/2]@ComPolar[{Mobius@(r Exp[I θ]),r Sin[θ]+2π n/4}]+{0,0,π},{n,0,4}]//ComplexExpand;

f=Table[Torus@Rot[0]@ComPolar2[{Unwrap@(r Exp[I θ]),r Sin[θ]+2π n/4}]+{0,0,π},{n,0,4}]//ComplexExpand;
ParametricPlot3D[f,{r,0,1},{θ,0,2π}]

ParametricPlot3D[Table[ComPolar[{r Exp[I θ],r Sin[θ]+2π n/4}],{n,0,4}],{r,0,1},{θ,0,2π}]

ParametricPlot[Unwrap[r Exp[I θ]],{r,0,1},{θ,0,2π},PlotPoints->100]

Unwrap[z_]:={Abs[z]-3,τArg[z]}
ComPolar2[{{a_,b_},f_}]:={a,b,f};

τArg[r Exp[I θ]]

Arg[0.1 Exp[I*-3]]