Wood – What tool should I use for carving deep shapes in a desk top

For an inexpensive and easy to work with material I would consider MDF. In particular there is coloured MDF available. The colour is throughout the board, not just painted on the surface, here's an example:

The benefit of this is that simply scratching the surface will not remove the colour. It also gives the MDF a neat texture as the colour is not 100% solid.

I'd make a template in the shape of a square donut out of 1/2" plywood by using a table saw to cut out the square hole in the 1/2" plywood. For example, if ...

i) the recess in the desktop needs to be 8"x12"

ii) the diameter of the base of the router is 6"

iii) the diameter of the router bit is 1/2"

... then I'd ...

1) start with a rectangular piece of plywood that measures 24"x28"

2) set the rip fence on the table saw to 5.25"

3) plunge cut on the table saw (advanced technique) along all four edges, but not all the way to the corners so as to leave the cutout solidly attached to the donut at all four corners.

4) make the final cuts in the corners with a jigsaw or handsaw or ...

At this point I ought to have a square donut template of the proper size. If the router is not high-end, then I'd mark a spot on the top of the router's base that I'd always keep rubbing against the template (this is important for precision work).

To route out the recess in the desktop, I'd secure the plywood template to the desktop. Then start excavating the recess at its center, progressing in an outward spiral until the final passes are done with the router's base bearing against plywood template.

Hope this helps.