References

It is perfectly reasonable to ask for a reference after 2 months. The internship is not going to last any longer, and it sounds like you had a good experience.

There is a difference between a reference and a letter of recommendation. A letter of recommendation is more detailed and specific, and it is more common for that to be asked for from someone who has known you for longer. For a reference, you can simply ask for a reference over email, or in person. You can say something like, "I really enjoyed my time here, and I would like to use you as a reference in the future. Would that be okay with you?"

I've been doing an internship that's to last 8 weeks in total, next week is my last week and whist I have been interning there for 2 months, it is reasonable of me to ask them for a reference that I can use in future?

I'm still an undergraduate student looking for other things to do but I'm not sure how to go about asking for a reference.

Yes, of course. They might have a policy to only confirm the job title and dates of employment in the reference, though.

I suggest you also download any payslips you have an e-mail them to your personal email as they will be useful for future background checks for other employers.

It is perfectly reasonable to ask for a reference after 2 months. The internship is not going to last any longer, and it sounds like you had a good experience.

There is a difference between a reference and a letter of recommendation. A letter of recommendation is more detailed and specific, and it is more common for that to be asked for from someone who has known you for longer. For a reference, you can simply ask for a reference over email, or in person. You can say something like, "I really enjoyed my time here, and I would like to use you as a reference in the future. Would that be okay with you?"