2 deleted 112 characters in body
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Husk, 55 4 bytes

Whoop, never got to use before (or ):

Szz▲TS‡▲T

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Explanation

Szz▲T  -- takes a matrix as argument, example: [[1,2],[3,4]]
S     T -- apply the followingfunction to itself ..
    T  -- .. and itself transposed:
 z   ‡▲  -- zip the two arguments with [[1,2],[3,4]] [[1,3],[2,4]]
  z▲   -bi- zip byvectorized maximum: [[1,3],[3,4]]

Husk, 5 bytes

Szz▲T

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Explanation

Szz▲T  -- takes a matrix as argument, example: [[1,2],[3,4]]
S      -- apply the following to itself ..
    T  -- .. and itself transposed:
 z     -- zip the two arguments with [[1,2],[3,4]] [[1,3],[2,4]]
  z▲   -- zip by maximum: [[1,3],[3,4]]

Husk, 5 4 bytes

Whoop, never got to use before (or ):

S‡▲T

Try it online!

Explanation

S  T -- apply the function to itself and itself transposed
 ‡▲  -- bi-vectorized maximum
1
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Husk, 5 bytes

Szz▲T

Try it online!

Explanation

Szz▲T  -- takes a matrix as argument, example: [[1,2],[3,4]]
S      -- apply the following to itself ..
    T  -- .. and itself transposed:
 z     -- zip the two arguments with [[1,2],[3,4]] [[1,3],[2,4]]
  z▲   -- zip by maximum: [[1,3],[3,4]]