In the most general sense, the Time Dependent Schrodinger Equation (TDSE) reads $$\hat{H} \Psi = i \hbar {{d} \over {dt}}\Psi $$

Is it possible to get rid of the $i$ entirely? Does it need to be there?

Let me be clear on the type of answer I am looking for. I am aware that the spatial wave functions can be written entirely real (by a theorem, whose name i do not recall), and that complex constants can be chosen such that the the time dependent part cancels out.

What I am instead asking is, is it possible to get an equivalent TDSE that does not deal with imaginary numbers in any sense? Or is it impossible because the $i$ is required for Hermitian operators?

In the most general sense, the Time Dependent Schrödinger Equation (TDSE) reads $$\hat{H} \Psi = i \hbar ~{{\mathrm d} \over {\mathrm dt}}\Psi $$

Is it possible to get rid of the $i$ entirely? Does it need to be there?

Let me be clear on the type of answer I am looking for. I am aware that the spatial wave functions can be written entirely real (by a theorem, whose name i do not recall), and that complex constants can be chosen such that the the time dependent part cancels out.

What I am instead asking is, is it possible to get an equivalent TDSE that does not deal with imaginary numbers in any sense? Or is it impossible because the $i$ is required for Hermitian operators?