I am trying to understand various ways in which one can modify the Langlands correspondence. Hopefully I will be able to learn something from you. First, one can categorify/decategorify.

It is my understanding the the geometric Langlands over complex numbers is by default categorical (yet there has been attempt to decategorify it by Frenkel and Langlands, see).

The arithmetic Langlands over finite extensions of $\mathbb{Q}_p$ is by default not categorical. I am not sure whether one can categorify it or not. Is there any literature on this?

For local fields of positive characteristic, I think the geometric Langlands and the arithmetic Langlands can co-exist (thus we have both a categorical statement and a non-categorical statement). Are there any implications from one to the other? A reference comparing the two in detail?

Second, one can also quantize. Gaitsgory is slowly but steadily understanding the "one-parameter" quantization of the geometric Langlands over complex numbers, it appears. There was a remark in Beilinson-Drinfeld about a certain "doubly quantized" local representation-theoretic result of Feigin-Frenkel, which they did not know how to globalize properly. This preprint appears to make first steps in that direction.

Can one also quantize or doubly quantize the usual arithmetic Langlands over finite extensions of $\mathbb{Q}_p$ or the function fields of positive characteristic? Can one quantize or doubly quantize the geometric Langlands in positive characteristic? Can one quantize or doubly quantize the de-categorified geometric Langlands over complex numbers? By the way, in the setting of Gaitsgory, the Langlands correspondence suddenly becomes more symmetrical than the non-quantized Langlands (i.e. you are computing "the same" categories, but with different parameters). Does something analogous happen here?

Possibly reasonable conjectures are hard to formulate for general reductive groups. What could the statements look like in the class field theory case, for example?

In all of the above variations, one can also ask how should the Arthur parameters (or the singular support condition, depending on what is more fundamental to you) look like. One can also ask how to incorporate ramification and how much ramification is to be allowed.

There is also one last way in which one can "modify" the Langlands correspondence but I am not sure it can even be properly called modification. In the local Langlands correspondence over $p$-adic numbers, you can either take the Galois representations with coefficients in $l$-adic numbers ($l \neq p$) or you can, in a pretty non-trivial fashion, take coefficients in $p$-adic numbers. What you get is the $p$-adic local Langlands correspondence. Can one $p$-adify any of the proposed correspondences above (where it makes sense)?

I realize that my questions are naive, I only wonder if it makes sense to ask them at all.