Just a curiosity that came up reading of this axiomatization of set theory: is $CH$ independent from Quine's New Foundations?
I can't seem to find anything on the topic. I thought that a possible reason for this lack of information is that the independence of $CH$ from $ZFC$ somehow trivially implies its independence from $NF$ and nobody thought it was something relevant to write on, but I don't know if this is the case.
We need to be careful to distinguish the relatively-well-understood theory $\mathsf{NFU}$ ($\mathsf{NF}$ + urelements) from its easier-to-motivate but surprisingly mysterious older sibling $\mathsf{NF}$.
There is a way to make sense of forcing for $\mathsf{NFU}$, and this results for example in a proof (following the usual lines) that $\mathsf{CH}$ is independent of $\mathsf{NFU}$. This is treated in Randall Holmes' paper Forcing in NFU and NF.
However, even "trivial" (from our $\mathsf{ZF}$-style experience) forcings will add ur-elements. So to get independence results over $\mathsf{NF}$ a further trick is needed; see Section 4 of Holmes' paper. This isn't too surprising, given that the consistency of $\mathsf{NF}$ (relative to, say, $\mathsf{ZFC}$) is still broadly open; Holmes has a claimed consistency proof, but my understanding is that it hasn't been vetted by the community yet.