For example, how to understand:

-i · h/(2·π) · ∂ψ/∂t = h^2/(8·π^2·m) · div grad ψ

(for simplicity in absence of potential multiplied by function).

The imaginary unit “i” simply shows that quantum operator is used instead of classical derivative, or function must be divided into two components: ψ = ψ1 + i · ψ2

and then in reality there are two equations

∂ψ1/∂t~div grad ψ2

∂ψ2/∂t~div grad ψ1

(~ symbol means is proportional with a constant multiplier).

In this case, the question arises how this relates to de Broglie equation, because it turns out to be

∂^2ψ1/∂t^2~div grad (div grad ψ1)

∂^2ψ2/∂t^2~div grad (div grad ψ2)

instead of traditional ∂^2ψ/∂t^2~div grad ψ

or ∂^2ψ/∂t^2~rot rot ψ for different kinds of waves.

Or is function real (should be, or can be)?

If Maxwell's equation is written as one formula, there are two components, electric field and magnetic, but instead of squared nabla single nabla (curl) is used, and this is consistent as de Broglie wave.

Do Pauli and Dirac equations follow the same principle as Schrödinger equation with respect to the complexity of function, or there are differences?