If you can tolerate solving for $1/m$ rather than $m$ (as $m=0$ doesn't given you real numbers), you could use the following:

    ContourPlot[(x^2 - y^2)/(x^2 + y^2)^2, {x, -2, 2}, {y, -2, 2},
     Contours -> {1, 1/2, 1/3, 1/4}, PlotRange -> All,
     ContourShading -> {White, Red, Blue, Green, Cyan}]

[![Contours][1]][1]

Maybe a more interesting display might be with `ContourPlot3D`:

    ContourPlot3D[(x^2 + y^2)^2 - m (x^2 - y^2), {x, -2, 2}, {y, -2, 2}, {m, 0, 4},
     Contours -> {0}, PlotRange -> All, PlotPoints -> 100]

[![ContourPlot3D example][2]][2]


  [1]: https://i.sstatic.net/LtI8y.png
  [2]: https://i.sstatic.net/DKTuU.png