I started studying conformal mappings some time ago, and made this visualisation that I would like to share. A conformal mapping locally preserves angles (i.e. its Jacobian is a scaled rotation matrix). In the figure below the black regular grid is mapped to the yellow warped grid. As you can see, all intersections of yellow lines are still at right angles (as they were in the black grid), even though the lines are themselves bent.
Conformal mappings are not linear, but they can be given a linear representation by embedding in Minkowski space with two additional dimensions. (This is similar to how homogeneous coordinates make translation and projection linear by increasing the dimension by one.) I recommend (Dorst, Fontijne, and Mann 2009) for more details.
Dorst, Leo, Daniel Fontijne, and Stephen Mann. 2009. Geometric Algebra for Computer Science. Revised Edition. Morgan Kaufmann.