It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$. The picture is less clear if $K$ is infinite-dimensional over $\mathbb{Q}$.

I believe the best result is that of Kobayashi who proved (modulo the usual conjectures on Hasse-Weil $L$-functions and BSD) that $\operatorname{rank}(E(\mathbb{Q}^{\operatorname{ab}}))=\infty$.

What is known about the rank of $E$ over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$?

EDIT: In view of Martin Bright's remark, let me point out that it would suffice for me if I could show that for every prime $p$ there are infinitely many primes $q$ so that:

(1) The rank of $E(\mathbb{Q}(\sqrt{-q}))$ equals that of $E(\mathbb{Q})$.

(2) $p$ has any prescribed splitting behavior (i.e. inert, split or ramified) in $\mathbb{Q}(\sqrt{-q})/\mathbb{Q}$.