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This is fairly simple one conceptually: just apply the conformal transformation $z \mapsto \frac{(2+3i)z-20}{(2-2i)z+12}$ to the square grid on the plane.

Here's the code (note: the If[] is to prevent polygons passing over infinity):

With[{a = 2 + 3 I, b = -20, c = 2 - 2 I, d = 12, 
  cols = RGBColor /@ {"#F2FCFC", "#0245A3"}},
 Manipulate[
  Graphics[
   Table[
    If[! (x == -3 && (y == -3 || y == -4 || y == -5)),
     {cols[[Mod[x + y, 2, 1]]],
      Polygon[
       Flatten[
        Table[ReIm[(a # + b)/(c # + d) &[t I + (x + y*I) + ((1 - s) E^( I θ) + s E^(I (θ + π/2)))]],
         {θ, π/4., 2 π, π/2}, {s, 0., 1, 1/10}], 
        1]]},
     Nothing],
    {x, -20, 12}, {y, -40, 5}],
   PlotRange -> {{0., 2.5}, {0., 2.5}}, ImageSize -> 540, 
   Background -> cols[[-1]]],
  {t, 0., 2}]]