That doesn't actually answer Douglas's statement that the continuum hypothesis is orthogonal to everything people care about if one assumes choice. In fact Doug's statement is more or less correct. See in particular discussion here. In particular, ZF + CH implies choice for sets of real numbers, which is what we care about for most practical purposes.
A comment at your link baldly asserts that ZF+CH implies choice for sets of real numbers, but the link seems otherwise irrelevant. Do you have a better citation? In particular, what do you mean by CH without choice? In fact, the comment asserts that ZF+CH implies R is well-orderable, which I don't think is true under weaker notions of CH.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.