We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$: $$ (-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0 $$ which defines the anti-commutator to be zero.
The requirement of SUSY charge $Q$ includes that
$Q$ is a Hermitian operator.
$[Q,H]=0$, $Q$ commutes with the Hamiltonian $H$ operator. $H$ is also Hermitian.
$Q^2$ is bounded from below. (Usually proportional to the Hamiltonian $H$ operator.)
Usually, in the literature, $Q$ is a linear and unitary operator. But can we have $Q$ to be instead antilinear and antiunitary?
My question is about the following, can we introduce a (new) SUSY charge called $Q'$ satisfy the additional less-common properties (other than satisfying the previous common properties mentioned above):
$Q'$ is an antilinear operator.
$Q'$ is an antiunitary operator.
Note that the (Hermitian) adjoint of the $Q'$ is also an antilinear and antiunitary operator. In fact, the (Hermitian) adjoint of the $Q'$ can be made to be the same $Q'$; thus $Q'$ can be regarded as Hermitian, or $Q'=Q'^\dagger$. See for example: https://physics.stackexchange.com/q/45227/12813.
Also, the product of two antilinear and antiunitary operators $Q'^2$ become a linear and unitary operator. Such as the complex conjugation (antilinear and antiunitary) $K$, whose square $K^2=+1$ is an identity (linear and unitary). Thus obeying conditions 4. and 5., do not seem to conflict with conditions 1.2.3. earlier.
Also, are there existing or previous literature introducing SUSY charge $Q'$ to be also antilinear and antiunitary?
