This may be a very naive question. We always hear about elliptic curves over the rational numbers, or over other arithmetically significant fields or rings. But, are there open problems or recent fertile theories related to elliptic curves over the complex numbers or is everything considered "classical" and well known?

For example, what about moduar parametrizations: are they only interesting as far as they involve curves defined over "arithmetic bases"? If yes, why?

What about elliptic cohomology, $\mathrm{tmf}$ and the like: if I'm not mistaken, the moduli spaces considered in that theory are defined over $\mathbb{Z}$; anything relevant/interesting happens over $\mathbb{C}$?

(I asked this question after having read this question -which, I must say, I'm not able to understand due to my ignorance of arithmetic geometry- in which one is lead to consider even points valued in -I think- de Rham differential forms...)