phi=(1+Sqrt[5])/2

sol=Solve[z==(1/ (phi I)) z + I + phi - 1/phi,{z}]//N

sol[[1,1,2]]

fixedpt=Simplify[(I  +1)/(1-(1/(phi I)))]//N

Clear[z]

Spiral[z_,n_]:=(1/(I phi))^n (z - fixedpt) + fixedpt

spiral1=Table[{Re[Spiral[0,.005t]],Im[Spiral[0,.005t]]},{t,0,4000}];

spiral1a=Table[{Re[Spiral[0,t]],Im[Spiral[0,t]]},{t,0,10}];

plot1=ListPlot[spiral1a,PlotStyle->PointSize[.015],PlotRange->{{-.1,phi+.1},{0,1.01}}]

plot2=ListPlot[spiral1a,PlotStyle->PointSize[.015],PlotRange->{{0,phi}+.01,{0,1.01}},Joined->True]

Show[plot1,plot2]

ListPlot[spiral1,PlotStyle->PointSize[.02],PlotRange->{{.5,phi}+.01,{0,1.01}},Joined->True]

.01

plot4 = ListPlot[spiral1,PlotStyle->PointSize[.02],PlotRange->{{.88,1.02},{.998,1.002}},Joined->True]

plot5 = Plot[1,{x,.88,1.02}]

Show[plot4,plot5]