Non standard extension of real numbers via nonprincipal ultra filters Assume  That $U,V$ are two  filters on the natural  number $\mathbb{N}$.
We say that $U$ is  equivalent to $V$ if there  is a  bijection $\phi: \mathbb{N} \to  \mathbb{N}$  such that $\tilde{\phi}(U)=V$ where $\tilde{\phi}:P(\mathbb{N}) \to  P(\mathbb{N})$ is  the  natural extension of  $\phi$ to the  power set $P(\mathbb{N})$.
Let $U,V$ be two  non principal ultra filter on $\mathbb{N}$.
Let $\mathbb{R}^*_{U}$ and $\mathbb{R}^*_{V}$ be the corresponding nonstandard extension of real numbers associated with $U$ and $V$, respectively.
Assume that  $\mathbb{R}^*_{U}$ and $\mathbb{R}^*_{V}$  are isomorphic as  fields. Does  this  imply that $U$ and  $V$ are  equivalent filters?
My  apology in advance, if the  question is  elementary. The  question arose me about  17 years  ago when I was  trying  to  understand the  application of  non standard  analysis  to ordinary  differential equations.
 A: As far as I understand the question is still open under $\neg CH$, namely whether isomorphism of hyperreal fields implies equivalence of filters (up to permutation of index set). Perhaps one can try the following approach. 
In a universe $V$ satisfying $CH$, we can take two inequivalent filters and obtain fields that are automatically isomorphic by the result of Erdos &Co mentioned in the MSE post linked in the comments above. Now the idea is to take a forcing extension $V^F$ satisfying $\neg CH$. 
One can't transfer naively the construction of the hyperreal field to $V^F$ because the ultrafilter is not definable, but perhaps one can work with the definable hyperreal field of Kanovei and Shelah (which exploits a huge index set using all ultrafilters simultaneously, thereby defeating non-definability). Perhaps one can specify two variants of the Kanovei-Shelah construction whether the ideals are not equivalent but the quotient fields will be by the Erdos-type argument, and then take a forcing extension to exhibit a similar phenomenon under $\neg CH$.
