I think you might find useful ImageTransformation and ImageForwardTransformation. I will crosspost here an answer from: How do I put an image on the complex plane? by J.M..

ImageForwardTransformation[] is the function you want here. To give a concrete example, here's how an image might be transformed by the complex mapping $w=z^3$:

img = ExampleData[{"TestImage", "Mandrill"}];
imgc = ImageForwardTransformation[img, Through[{Re, Im}[(#[[1]] + I #[[2]])^3]] &,
                         Background -> 1,
                         DataRange -> {{-1, 1}, {-1, 1}}, PlotRange -> {{-2, 2}, {-2, 2}}]

To see the correspondence with the more usual complex mapping, we show the transformed image along with a suitably transformed Cartesian grid:

ParametricPlot[{Re[(x + I y)^3], Im[(x + I y)^3]}, {x, -1, 1}, {y, -1, 1},
               PlotStyle -> FaceForm[None], Prolog -> {Texture[imgc],
               Polygon[Scaled /@ {{0, 0}, {1, 0}, {1, 1}, {0, 1}},
                       VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}]}]

As an example of a nontrivial complex mapping, here is the conformal mapping of a square region to a disk:

img = ExampleData[{"TestImage", "Mandrill"}];
imgc = With[{? = N[1/2 EllipticK[1/2], 25]},
            ImageForwardTransformation[img, 
                         With[{z = ? (#[[1]] + I #[[2]])},
                              Through[{Re, Im}[JacobiSC[z, 1/2] JacobiDN[z, 1/2]]]] &, 
                         Background -> 1, DataRange -> {{-1, 1}, {-1, 1}}, 
                         PlotRange -> {{-1, 1}, {-1, 1}}]]

Another nontrivial example of a complex mapping (the quincuncial projection) is demonstrated in this answer (though the procedure given there uses ImageTransformation[] instead).