My question is about a specific example of supersymmetry in quantum mechanics. I am not an expert on SUSY, and I would like to have some insights on this. Imagine you have a non-Hermitian supercharge $Q$ satisfying the algebra \begin{equation} \{ P, Q \}=0,\,\,\,\, Q^2=( Q^\dagger)^2=0, \,\,\, \{Q,Q^\dagger\}=H, \end{equation} where $P$ is the parity operator and $H$ is the Hamiltonian.
To be specific, let's assume that $H$ is a quantum mechanical Hamiltonian (e.g., Schrödinger) in one dimension.
In this case, since the supercharge is not-Hermitian and can be written as a sum of two Hermitian operators, one says that the Hamiltonian has $\mathcal N=2$ supersymmetry.
Imagine now that one has a second supercharge $Q'$ such that $[Q,Q']\neq0$ and $\{Q,Q'\}\neq0$, satisfying the same algebra \begin{equation} \{ P, Q' \}=0,\,\,\,\, Q^{\prime 2}=(Q^{\prime\dagger})^2=0, \,\,\, \{Q',Q^{\prime\dagger}\}=H, \end{equation}
My question(s): In this case, one is talking about $\mathcal N=4$, $\mathcal N=(2,2)$ or else? What is the physical significance of supersymmetry with respect to two independent supercharges?
