calculating variations – Optimizing a complex functional compared to the lexicographic command?

I wonder if the following argument is correct:

Consider optimizing a complex functional S[x(t)]. Since S is complex, it has only one optimum vis-à-vis the lexicographic order of complex numbers. (This is different from the stability of Cauchy-Riemann, where δS = 0, but there is no notion of optimality).

To optimize the lexicographic order of complex numbers, we first optimize Real (S)

0 = δ (real S)

Then, we optimize the imaginary (S) while keeping δ (Real S) = 0.

This is the part of which I am not sure: the Lagrange multiplier method is not appropriate here, and in general, it can not be said that the optimization of Imaginary S will also stabilize it . However, it can be said that the difference between stabilizing Imaginary S and optimizing it is an unpredictable noise, equal to 0 on average.

δ (S imaginary) = noise

Adding them gives δS = i noise.

Does my argument that δ (Imaginary S) = noise is correct? Moreover, what can we say about this noise?