I'm not on MO often - funny I stumbled on your question... In general, as you noticed, the analogous statement to Mazur's theorem is false: an example is $\mathbb{Q}[i]$, where the curve $j=1728$ has a rational isogeny of order $p$ for any prime $p=1$ mod $4$.

However, you can still hope for it to hold for fields $K$ which don't admit any CM curves defined over $K$ whose CM field is contained in $K$ (for example this holds for all but finitely many quadratic imaginary $K$ - namely, all except those with class number $1$.)

For $K$ satisfying this hypothesis, I'm pretty sure what you asked is in fact true: there is a finite list of primes p for which a p-isogeny is defined (in fact there's even a finite list of prime powers q with cyclic q-isogenies.) I don't think there's a proof in the literature, but I'm actually writing up a proof of this fact currently with Eric Larson. Our proof uses Galois representations instead of counting points on modular curves and so is very different from Mazur's (and the bounds for p it gives are very big for most fields K). A preprint should be up within the next couple of months.

If you're really interested in a field $K$ that admits such CM curves, you can ask whether there's a finite list of primes that can be isogenies of a non-CM curve $E$. This question is harder, since you'd need some way of distinguishing CM from non-CM curves, or at least carefully counting points on $X_0(N)$, and it is certainly still open. However hypothetically this is still true (it falls under the more general hypothesis of Serre that there's a finite list of "exceptional" primes of non-CM curves over a number field K.)

I'm glad to have stumbled on your question -- I'm actually working on something along the lines of what you're asking about now with Eric Larson. If what we think we've proven is true (and don't quote this yet, since we're not yet even done with the write-up,) then in fact you can say something stronger: you cannot have isogenies of order p for any prime p sufficiently large (where "sufficiently large" depends on K), as long as K does not contain the class field of an imaginary quadratic extension in which p splits (this exactly gets rid of primes which are isogenies of a CM curve defined and with CM over K). So this would mean that if K is quadratic imaginary, there can be no p-isogenies for any p sufficiently large unless K is one of the finitely many quadratic imaginary fields of class number one (in which case, if our arguments work, this would mean that all but finitely many possible prime degrees of isogeny p split over K.)

If you're interested in what happens to primes that split, you can ask whether, more generally, for any number field K, there are only finitely many primes that can be isogenies for a curve over K without CM. This seems harder to prove, and as far as we know, is an open problem (to prove it, you'd need to have a good way of distinguishing CM from non-CM curves, or carefully counting points on $X_0(N)$). But hypothetically this is also true: it fits into the general framework of Serre's hypothesis that "exceptional primes" of curves without CM are bounded by a constant depending only on $K$.

I'm not on MO often - funny I stumbled on your question... In general the analogous statement is false: an example is $\mathbb{Q}[i]$, where the curve $j=1728$ has a rational isogeny of order $p$ for any prime $p=1$ mod $4$.

However, you can still ask your question for fields $K$ which don't admit any CM curves defined over $K$ whose CM field is contained in $K$ (this exactly gets rid of examples like $\mathbb{Q}[i]$ where there's a CM counterexample to your question. It's true for example for any imaginary quadratic field of class number $\ne 1$ - hypothetically all but finitely many.)

In this case, I'm pretty sure what you asked is in fact true: there is a finite list of primes p for which a p-isogeny is defined (in fact there's even a finite list of prime powers q with cyclic q-isogenies.) I don't think there's a proof in the literature, but I'm actually writing up a proof of this fact currently with Eric Larson. Our proof uses Galois representations instead of counting points on modular curves and so is very different from Mazur's (and the bounds for p it gives are very big for most fields K). A preprint should be up within the next couple of months.

If you're really interested in a field $K$ that admits such CM curves, you can ask whether there's a finite list of primes that can be isogenies of a non-CM curve $E$. This question is harder, since you'd need some way of distinguishing CM from non-CM curves, or at least carefully counting points on $X_0(N)$, and it is certainly still open. However hypothetically this is still true (it falls under the more general hypothesis of Serre that there's a finite list of "exceptional" primes of non-CM curves over a number field K.)

I'm not on MO often - funny I stumbled on your question... In general, as you noticed, the analogous statement to Mazur's theorem is false: an example is $\mathbb{Q}[i]$, where the curve $j=1728$ has a rational isogeny of order $p$ for any prime $p=1$ mod $4$.

However, you can still hope for it to hold for fields $K$ which don't admit any CM curves defined over $K$ whose CM field is contained in $K$ (for example this holds for all but finitely many quadratic imaginary $K$ - namely, all except those with class number $1$.)

For $K$ satisfying this hypothesis, I'm pretty sure what you asked is in fact true: there is a finite list of primes p for which a p-isogeny is defined (in fact there's even a finite list of prime powers q with cyclic q-isogenies.) I don't think there's a proof in the literature, but I'm actually writing up a proof of this fact currently with Eric Larson. Our proof uses Galois representations instead of counting points on modular curves and so is very different from Mazur's (and the bounds for p it gives are very big for most fields K). A preprint should be up within the next couple of months.

If you're really interested in a field $K$ that admits such CM curves, you can ask whether there's a finite list of primes that can be isogenies of a non-CM curve $E$. This question is harder, since you'd need some way of distinguishing CM from non-CM curves, or at least carefully counting points on $X_0(N)$, and it is certainly still open. However hypothetically this is still true (it falls under the more general hypothesis of Serre that there's a finite list of "exceptional" primes of non-CM curves over a number field K.)