You have, 

    gamma1[xi1_, xi2_] = (0. - 1.72945 I) e2 (-0.3567 xi1 + 0.229 xi2 + 
           0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 - (0. + 
          0.5 I) (eps (0.5929 xi1 - 0.3806 xi2 - 
             1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 
          38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 
              0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);
    gamma2[xi1_, xi2_] = (-1.72945 + 0. I) e2 (-0.3567 xi1 + 0.229 xi2 + 
           0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3 + (0.5 + 
          0. I) (eps (0.5929 xi1 - 0.3806 xi2 - 
             1.2296 (-0.0423 eps xi1 + 0.4227 eps xi2)) + 
          38.4563 e2 (-0.3567 xi1 + 0.229 xi2 + 
              0.7398 (-0.0423 eps xi1 + 0.4227 eps xi2))^3);

Just rewrite your variables in terms of `z` and `zc`, 
    
    Solve[{z == xi1 + I xi2, zc == xi1 - I xi2}, {xi1, xi2}]
    (* {{xi1 -> (z + zc)/2, xi2 -> -(1/2) I (z - zc)}} *)

Then substitute them into the original expression:
    
    
    gamma1[(z + zc)/2, -(1/2) I (z - zc)] // FullSimplify
    gamma2[(z + zc)/2, -(1/2) I (z - zc)] // FullSimplify
    (* (0.00641225 - 0.00197804 I) e2 ((0.838054 - 1.0568 I) z + 
        1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
           0.198158 I) eps zc)^3 - (0. + 
        0.5 I) (eps (((0.29645 + 
                0.1903 I) + (0.026006 + 0.259876 I) eps) z + ((0.29645 - 
                0.1903 I) + (0.026006 - 
                 0.259876 I) eps) zc) + (0.043984 + 
           0.142584 I) e2 ((0.838054 - 1.0568 I) z + 
           1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
              0.198158 I) eps zc)^3) *)
    (* (-0.00197804 - 0.00641225 I) e2 ((0.838054 - 1.0568 I) z + 
        1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
           0.198158 I) eps zc)^3 + (0.5 + 
        0. I) (eps (((0.29645 + 
                0.1903 I) + (0.026006 + 0.259876 I) eps) z + ((0.29645 - 
                0.1903 I) + (0.026006 - 
                 0.259876 I) eps) zc) + (0.043984 + 
           0.142584 I) e2 ((0.838054 - 1.0568 I) z + 
           1. eps z - (0.612024 + 1.20191 I) zc - (0.98017 + 
              0.198158 I) eps zc)^3) *)