`Part` like `Delete`: How to delete list of columns or arbitrarily deeper levels

You can actually use Part for that:

ClearAll[delete];
delete[expr_, specs___] :=
  Module[{copy = expr},
    copy[[specs]] = Sequence[];
    copy
  ];

So that for example

delete[m, All, All, 2]

(* 
   {
     {{1, 3}, {4, 6}, {7, 9}, {10, 12}}, 
     {{13, 15}, {16, 18}, {19, 21}, {22, 24}}, 
     {{25, 27}, {28, 30}, {31, 33}, {34, 36}}, 
     {{37, 39}, {40, 42}, {43, 45}, {46, 48}}
   }
*)

Note that this is not exactly equivalent to Delete in all cases, since sequence splicing is an evaluation-time effect, so the results will be different if you delete inside held expressions - in which case the method I suggested may not work.

Here is a version that would probably be free of the mentioned flaw, but will be slower:

ClearAll[deleteAlt];
deleteAlt[expr_, specs___] :=
  Module[{copy = Hold[expr], tag},
    copy[[1, specs]] = tag;
    ReleaseHold@Delete[copy, Position[copy, tag]]
  ];

You can test both on say, Hold[Evaluate[m]], with the spec 1, All, All, 2, to see the difference.

You may use ReplacePart and Nothing with Blank (_) for All.

Delete second entry of all submatrices

ReplacePart[m, {_, _, 2} -> Nothing] // MatrixForm

Delete last column

ReplacePart[m, {_, -1} -> Nothing] // MatrixForm

and so on.

Hope this helps.