Does the quantum evolution equation has to be a complex first order equation?

The problem of equation (1) is the difficulty of introducing probability. In regular quantum mechanics one of the ways to get probability is from the symmetry of the effective Lagrangian $$ L=ipsi^{dagger}partial_t psi-psi^{dagger}hat{H}psi. $$ Here $psi=(psi_1,psi_2,....)$ and $psi^{dagger}=(psi^*_1,psi^*_2,....)$, where $psi^*_1, psi^*_2,... $ are new auxiliary variables, independent of $psi_1,psi_2,...$. Under the symmetry $psirightarrowpsi e^{ialpha}$, $psi^{dagger}rightarrowpsi^{dagger} e^{-ialpha}$ such Lagrangian generates a conserved quantity

$$Q=psi^daggerpsi,$$ which we associate with probability.

Doing a similar trick with the Eq. (1) yields a very different result. The Lagrangian in this case is $$ L=partial_tpsi^{dagger}partial_t psi-psi^{dagger}hat{H}^2psi. $$ The corresponding conserved quantity due to the phase symmetry is $$ Q=i(psi^{dagger}partial_t psi-partial_t psi^{dagger} psi), tag{2} $$ which can be positive or negative. What is worse, for all real $psi$ it is zero. In other words, $Q$ cannot be probability!

In the comments to the OP there was a suggestion that since the equation of motion (1) corresponds to a collection of coupled oscillators, may be the "energy" of such system (kinetic+potential), $$ Q=partial_t psi^{dagger}partial_tpsi+ psi^{dagger}hat{H}^2psi, $$ could be associated with probability. The problem with this attempt is that if Hamiltonian $hat{H}$ depends on time, the "energy" (and therefore the corresponding probability) will not be conserved, which, in turn, would make non-local evolution of probability possible.