Alternatively shift columns and rows of a 2D array

MATL, 13 bytes

,!tZy:oEq2&YS

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Explanation

,        % Do twice
  !      %   Transpose. Takes input implicitly the first time
  t      %   Duplicate
  Zy     %   Size. Gives a vector with numbers of rows and of columns
  :      %   Range from 1 to the first entry of the vector (number of rows)
  o      %   Parity: gives 0 or 1 for eacn entry
  Eq     %   Times 2, minus 1: transforms 0 into -1
  2      %   Push 2
  &YS    %   Circularly shift along the second dimension. This shifts the
         %   first row by 1 (that is, to the right), the second by -1 (to
         %   the left), etc.
         % End (implicit). Display (implicit)

J, 26, 21 19 bytes

-5 bytes thanks to miles

(|."_1~_1^#\)@|:^:2

Explanation:

^:2 - repeate twice the following:

@|: - transpose and

#\ - find the length ot the prefixes (1, 2, 3 ... rows)

_1^ - raise -1 to the above powers, creating a list of alternating -1 1 -1 1...

|."_1~ - rotate each row of the input array with offset from the above list

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Original version:

(($_1 1"0)@#|."0 1])@|:^:2

How it works

^:2 - repeate twice the following:

|: - transpose and

|."0 1] - rotate each row of the input array, offsets in the list:

@# - the number of rows in the array

($_1 1"0) - alternate _1 1 (3 -> _1 1 _1)

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Wolfram Language (Mathematica), 55 bytes

Nest[MapIndexed[RotateLeft[#,(-1)^#2]&,Thread@#]&,#,2]&

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