Lagrangian density for a complex scalar field

I am taking a course on classical field theory, and am not entirely sure as to what motivates the for of the Lagrangian density for a complex scalar field. In my lecture notes, this is first introduced in the context of the Klein-Gordon equation, and the following statements are made:

1. $psi$ can be decomposed into $psi = psi_1 +ipsi_2$ where $psi_1$ and $psi_2$ are independent. Thus $psi$ and $psi^*$ are independent.

2. The Lagrangian density $L=L[psi_1]+L[psi_2]$ can be written $L=frac{partial psi^*}{partial t}frac{partial psi}{partial t}-frac{partial psi^*}{partial x}frac{partial psi}{partial x} – m^2psi^*psi$.

Now I am okay with the consistency of point 2 with itself. I.e. I see that The form of the Klein-Gordan Lagrangian density given does indeed give $L=L[psi_1]+L[psi_2]$, and conversely writing the Klein Gordon Lagrangian as $L=L[psi_1]+L[psi_2]=L[psi + psi^*]+L[psi-psi^*]$ gives the explicit form form.

However I am not at all sure why the Lagrangian density is made to be additive in $psi_1$ and $psi_2$. I do not see the motivation for this, other than that it reproduces the Euler-Lagrange equations for each of the independent scalar fields separately. Is there any better explanation?