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](https://en.wikipedia.org/wiki/Non-standard_analysis#Basic_definitions) 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.



Non standard extension of real numbers via nonprincipal ultra filters
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 $\neg CH$.


