Let K denote either the field of real numbers or the complex fields. An analytic function over $K^n$ is a function that can be represented locally by a convergent power series in n variables with coefficients in K.


My question is that can we take K to be other fields? It seems that such a field K should satisfy some criteria:


1. It is a metric space or at least a topological space.
2. There should be complete, in the sense that Cauchy nets converge.
3. The above 2 points probably force that K should have cardinality at least the size of the continuum.


Can there be other K where a reasonable theory of analytic functions can be developed? Say for cardinalities larger than the continuum? Probably this invokes some model theory. 



Analytic Functions over Fields other than Real or Complex Numbers
There are several rich theories of analysis on non-archimedian theories. Neal Koblitz' book on $p$-adic analysis is a good introduction. Non-archimedian analysis by Bosch, Güntzer and Remmert is more encyclopedic. Berkovich's Spectral Theory and Analysis over Non-archimedian Fields introduces his beautiful theory of analytic spaces allowing for a reasonable algebraic topological theory. In Goss's book Basic Structures of Function Field Arithmetic there is a good introduction to analysis in positive characteristic.


Your suggestion that this subject might have something to do with model theory is apt. As the above references show, the theory may be developed without model theory, but it has been studied intensively via model theory giving interesting results about quantifier elimination, uniformity across the $p$-adics, and establishing a basis for motivic integration. You might want to look at the paper by van den Dries and Denef, $p$-adic and real subanalytic sets. Ann. of Math. (2) 128 (1988), no. 1, 79--183.


