I tried to solve the quantum harmonic oscillator via the operator method. After doing it and looking up the solution I noticed that for some reason the ladder operators got an additional factor of (i) / (-i) in them. Is this just a (weird) convention? In my calculations I don't seem to need it and since only a product appears in the Hamiltonian it does not matter if there is an (i)*(-i)=1 in there or not.
My calculations are made with pencil - the extra (i) & (-i) is in blue.