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In[]:=

ClearAll["Global`*"]

b1=FullSimplify[Arg[z]+Arg[Conjugate[z]],Element[z,Complexes]&&Im[z]!=0]

arg[z_]:=Module[{x=Re[z],y=Im[z],r},r=Abs[z];If[y!=0,2ArcTan[y/(r+x)],If[x>=0,0,Pi]]];b2=Simplify[arg[z]+arg[Conjugate[z]],Element[z,Complexes]&&Im[z]!=0]