The log of a negative number is an imaginary number. You can change the sign of the real parts that are negative as follows:
Eps1[\[Omega]_] := (p[\[Omega]][[1]]^2 - p[\[Omega]][[2]]^2) +
I*2*(p[\[Omega]][[1]]*p[\[Omega]][[2]]);
LogPlot[{-Re[Eps1[\[Omega]]], Im[Eps1[\[Omega]]]}, {\[Omega], 1, 5},
PlotStyle -> Thick, PlotRange -> All]