Assume that $A$ and $B$ are two dependent random variables. Are there any results on functions $f$ such that 
$C =f(A, B)$ and $A$ are independent? 


For example, it can easily be shown that $A$ and $C = F\_{B|A}(B, A)$ are independent where $F\_{B|A}(., .)$ is the conditional CDF function of $B$, given $A$. (I am using the conditional CDF as a function of two random variables). This is called the Darmois decomposition in the signal processing literature.


So $A$ and $h\Big(F\_{B|A}(B, A)\Big)$ are also independent where $h$ is a measurable function. 


Are there any other functions such that $C =f(A, B)$ and $A$ are independent? 



Functions $f$ such that $A$ and $f(A,B)$ are independent
I must say that I don't quite understand your notation. There is a complete description of the independent complements in the sense you are asking, but I prefer to formulate it in somewhat different terms, namely in the language of Lebesgue spaces and their measurable partitions due to [Rokhlin](https://mathscinet.ams.org/mathscinet-getitem?mr=30584). The base probability space in this problem is the joint distribution of $A$ and $B$; let us denote this space by $(X,m)$. Then the random variable $A$ is a function on the space $X$. By one of Rokhlin's theorems, if the distribution of $A$ and almost all conditional distributions of $B$ are purely non-atomic, then the space $(X,m)$ can be identified with the unit square endowed with the Lebesgue measure, and $A$ with the projection of this square onto the first coordinate. Then in this setup you are asking about functions $C$ on the unit square endowed with the Lebesgue measure which are independent of the first coordinate. They are all of the form
$$
C(x\_1,x\_2) = \phi ( \Psi^{x\_1} (x\_2)) \;,
$$
where $\Psi^{x\_1}$ is a measurable family of automorphisms of the unit interval, and $\phi$ is a measurable function on the unit interval.


