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We use ReIm@p the get the real pairs then use FindFit.

sol = FindFit[ReIm@p, a x + b, {a, b}, x];
eqnR[x_] := a x + b /. sol;
eqnR[x]
Show[ListPlot[ReIm@p, Joined -> False], 
 Plot[eqnR[x], {x, 0, 1.4}, PlotStyle -> Red]]