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Like this:

sage: A = ZZ^4; B = ZZ[I]
sage: H = Hom(A,B)
sage: f = H( [B.one(), -I, -I, B.one()] )
sage: f
Free module morphism defined by the matrix
[ 1  0]
[ 0 -1]
[ 0 -1]
[ 1  0]
Domain: Ambient free module of rank 4 over the principal ideal domain Integer Ring
Codomain: Gaussian Integers generated by I0 in Number Field in I0 with defining polynomial x^2 + 1 with I0 = 1*I
sage: f.kernel()
Free module of degree 4 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1  0  0 -1]
[ 0  1 -1  0]

Like this:

sage: A = ZZ^4; B = ZZ[I]
sage: H = Hom(A,B)
sage: f = H( [B.one(), -I, -I, I, B.one()] )
sage: f
Free module morphism defined by the matrix
[ 1  0]
[ 0 -1]
[ 0 -1]
 1]
[ 1  0]
Domain: Ambient free module of rank 4 over the principal ideal domain Integer Ring
Codomain: Gaussian Integers generated by I0 in Number Field in I0 with defining polynomial x^2 + 1 with I0 = 1*I
sage: f.kernel()
Free module of degree 4 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1  0  0 -1]
[ 0  1 -1  1  0]

Like this:

sage: A = ZZ^4; B = ZZ[I]
sage: H = Hom(A,B)
sage: f = H( [B.one(), -I, I, B.one()] )
sage: ); f
Free module morphism defined by the matrix
[ 1  0]
[ 0 -1]
[ 0  1]
[ 1  0]
Domain: Ambient free module of rank 4 over the principal ideal domain Integer Ring
Codomain: Gaussian Integers generated by I0 in Number Field in I0 with defining polynomial x^2 + 1 with I0 = 1*I
sage: f.kernel()
Free module of degree 4 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1  0  0 -1]
[ 0  1  1  0]