We study open symplectic manifolds with pseudoholomorphic $\mathbb C^*$-actions whose $S^1$-part is Hamiltonian, and construct their associated symplectic cohomology. From this construction, we obtain a filtration on quantum/ordinary cohomology that depends on the choice of the $\mathbb C^*$-action. One should think about this filtration as a Floer-theoretic analogue of the Atiyah-Bott filtration. We construct filtration functional on the Floer chain complex, allowing us to compute the aforementioned filtration via Morse-Bott spectral sequence that converges to symplectic cohomology, which is readily computable in examples. We compare our filtration with known ones from algebraic geometry/representation theory literature. Time-allowing, I may present the $S^1$-equivariant picture as well. This is joint work with Alexander Ritter.