M = {{1,1,1,1},{2,2,4,4},{3,3,5,5},{9,9,3,1}}

M = {{1,1,1,1},{2,2,4,4},{3,3,5,5},{9,9,3,3}}

M = {{1, 1, 1, 1}, {2, 2, 4, 4}, {3, 3, 5, 5}, {9, 9, 3, 1}};

Select[GatherBy[Range[Length@M], Transpose[M][[#]] &], Length[#] > 1 &]

M2 = {{1, 1, 1, 1}, {2, 2, 4, 4}, {3, 3, 5, 5}, {9, 9, 3, 3}};

Select[GatherBy[Range[Length@M2], Transpose[M2][[#]] &], Length[#] > 1 &]