We describe certain partial results (some with M. Weber) about boundedness of ergodic square functions for subsequence and generalised averages -- like Bourgain averages and Templeman averages. These results and many natural and associated questions complement the very complete theory for Birkhoff averages associated with names like Ackoglu, Jones, Kaufman, Ostrowski, Rosenblatt, Schwartz, Wierdl.