Are there any analogues of Galois Theory for complex numbers (with non-zero imaginary part)? This is motivated by Schwarz Principle. It says that if $f$ is analytic, $f$ is defined in the upper-half disk, and $f$ extends to a continuous function on the real axis, then $f$ can be extended to an analytic function on the whole disk by the formula $f(\bar{z}) = \overline{f(z)}$.
This reminds me of the Isomorphism Extension Theorem.
Answer
Galois theory is useful when you have some algebraic object, and a list of tools you are allowed to use within that object. The purpose of Galois theory is to explain how far one can go only using those tools.
For example, it is impossible to create, using only the tools of +, -, *, / and nth roots, a formula for the zeroes of a general 5th degree polynomial. This is the most basic application of Galois theory. This is the Abel-Ruffini theorem:
http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
Another application is Differential Galois theory, which tells you which integrals can be expressed in terms of elementary functions. But once again, it comes down to having only a few basic tools with which to construct an anti-derivative for a function using only elementary functions, and that is why Galois theory is useful in this application.
So, its hard to understand what your question is asking about, because you haven't really phrased your question as a Galois Theory question.
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