R, 121121 94 bytes

function(A)t(outer(1:(n=NROW(A)),1:n,Vectorize(function(i,j)(-1)^(i+j)*det(if(is.matrix(a<-A[-i,-j]))a else matrix(a)j,drop=F]))))

This is an absurdly long function that accepts an object of class matrix and returns another such object. To call it, assign it to a variable.

Ungolfed:

cofactor <- function(A) {
    # Get the number of rows (and columns, since A is guaranteed to
    # be square) of the input matrix A
    n <- NROW(A)

    # Define a function that accepts two indices i,j and returns the
    # i,j cofactor
    C <- function(i, j) {
        # Since R loves to drop things into lower dimensions whenever
        # possible, ensure that the minor obtained by column deletion
        # is still a matrix object by adding the drop = FALSE option
        a <- A[-i, -j]
        if (!is.matrix(a))j, adrop <-= matrix(a)FALSE]

        (-1)^(i+j) * det(a)
    }

    # Obtain the adjugate matrix by vectorizing the function C over
    # the indices of A
    adj <- outer(1:n, 1:n, Vectorize(C))

    # Transpose to obtain the cofactor matrix
    t(adj)
}