We use `ReIm@p` to 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]]
```
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/ymBxD.png