Here is a code that plots your arrows correctly:

    Cases[1.56664 DiracDelta[(4.3 + 0. I) - x] + 
       I Sqrt[\[Pi]/2] DiracDelta[(4.6 + 0. I) - x] + 
       1.56664 DiracDelta[(4.3 + 0. I) + x] - 
       I Sqrt[\[Pi]/2] DiracDelta[(4.6 + 0. I) + x], 
      a_ DiracDelta[b_] :> {{a}, x /. Solve[b == 0]}];
    di = Flatten[Tuples /@ %, 1];
    im = Select[di, Im[#[[1]]] != 0 &];
    re = Select[di, Im[#[[1]]] == 0 &];
    mm = Max[Abs[Join[re[[All, 1]], Im[im[[All, 1]]]]]] + 0.1;
    Plot[{}, {x, -5, 5}, 
     Epilog -> {Red, Arrow[{{#[[2]], 0}, {#[[2]], #[[1]]}}] & /@ re, Blue,
        Arrow[{{#[[2]], 0}, {#[[2]], Im[ #[[1]]]}}] & /@ im}, 
     PlotRange -> mm]
    Clear[di, im, re, mm]

[![enter image description here][1]][1]


He are more complex `DiracDelta` functions plotted:

    -2 DiracDelta[x + 1] + 5 DiracDelta[(7 x + 3) (x - 4)] - 
     3 I DiracDelta[x - 3/2] + 2 I DiracDelta[x + 3]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/dIomK.png
  [2]: https://i.sstatic.net/IRvIn.png