It is consistent that all ultrafilters are the same in the following sense: *for any free ultrafilters $\mathcal U,\mathcal V$ on $\omega$ there exists a finite-to-one map $f:\omega\to\omega$ such that $f(\mathcal U)=f(\mathcal V)$*. The latter statement is called NCF, the Near Coherence of Filters. In some situation it is very helpful principle. It has been thoroughly studied by Andreas Blass, see his papers: [I][1], [II][2], [III][3].


  [1]: https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-27/issue-4/Near-coherence-of-filters-I-Cofinal-equivalence-of-models-of/10.1305/ndjfl/1093636772.full "Near coherence of filters. I. Cofinal equivalence of models of arithmetic."
  [2]: https://www.ams.org/journals/tran/1987-300-02/S0002-9947-1987-0876466-8/S0002-9947-1987-0876466-8.pdf "Near Coherence of Filters. II: Applications to Operator Ideals, the Stone-Čech Remainder of a Half-Line, Order Ideals of Sequences, and Slenderness of Groups"
  [3]: https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-30/issue-4/Near-coherence-of-filters-III-A-simplified-consistency-proof/10.1305/ndjfl/1093635236.full "Near coherence of filters. III. A simplified consistency proof."