For a map from the plane to itself that may be described by a complex-valued function of a complex variable, the problem has already been solved by David Parks in his Presentations add-on. Specifically, the ComplexMap function there does what you ask. For example:

<< Presentations`

(* objects in domain to be transformed *)
objects = {
  (* two points *)
  Red, PointSize[Large], ComplexPoint /@ {1 + I, 1 - I},
 (* parameterized vertical segment *)
  ComplexCurve[(1 - s) (1 + I) + s (1 - I), {s, 0, 1}, PlotStyle -> Directive[Dashed, Green]],
 (* parameterized parabolic arc *)
  ComplexCurve[t^2 + t I, {t, -1, 1}, PlotStyle -> Directive[Thick, Blue]]
};

(* draw the objects in the complex domain *)
dom = Draw2D[{objects}, Axes -> True, PlotRange -> 1.5, ImageSize -> Scaled[0.4]];

(* the transformation *)
f[z_] := z^2

(* draw the images of the objects under f *)
img = Draw2D[{objects // ComplexMap[f]}, Axes -> True, PlotRange -> 2.25, ImageSize -> Scaled[0.4]];

(* display both domain and codomain *)
Row[{dom, Spacer[10], img}]

Naturally, the "dirty work" is done behind-the-scenes by ComplexMap, defined in the add-on. By careful analysis of the definition of ComplexMap, you could create an analog directly in terms of Cartesian coordinates that would work for any transformation of the form {u,v} = f[{x, y}].

Mathematica Graphics are a vector graphics language. And most vector graphics are made up of polygons and stuff like that. Arbitrary transforms don't map polygons to polygons.

You can do this with an Image though... to some extent: http://reference.wolfram.com/language/ref/ImageTransformation.html