I am familiar with the theory of modular forms and weight k Eisenstein series, and I am wondering if such a theory exists when the base field is not $\mathbb{Z}$. 


Is there a theory of modular forms over $SL\_2(\mathcal{O\_k})$ where $k$ is a real or imaginary quadratic number field? Moreover, is there a nice Fourier series expansion of their Eisenstein series (if they exist)? Please provide references. 



Eisenstein series for quadratic number fields
When $k$ is a real quadratic field (or more generally a totally real number field) the short answer is Hilbert modular forms. The corresponding Eisenstein series are called Hecke-Eisenstein series and are quite easily defined,
and yes they have a very nice Fourier expansion. For instance, this is what allowed Siegel to show that $\zeta\_K(1-k)$ is always a rational number for
$k\ge2$.


The case of imaginary quadratic fields is a little more complicated: the modular forms are here called Bianchi modular forms.


There is a vast literature on Hilbert mf, a little less on Bianchi.
If I may, you can look at the last chapter of my recent book with Fredrik Str\"omberg published by the AMS.


