Yes, you can do it.

In fact, In near future I too want to continue my research independently and publish my results.

However, there are few things you need to remember before doing that.

First,

It will be useful if you are familiar with the research methods and publishing ways. You see even if you have some brilliant discovery, your idea will be only appreciated by scientific community if you can convey in a right way.

For example, even in my undergraduate studies, I used to have many interesting ideas. But I did not know how to publish them.

Second,

There is a reason why people tend to do research affiliated with some college or organization i.e. Funding/Resources. It is hard to conduct an individual research as there can be many expenses required in the process.

For example, you wanted to simulate a physical system and the results are integral part of your research. But if you simulate it on a desktop, it will take years to complete. So you need a HPC cluster, which is too costly and not feasible for everyone to have it.

That is the reason why researchers tend to collaborate with colleges or institutions.

However, there is a positive note on this decision, Freedom

You will have all the time in the world, and you can do it anytime you want. There will be no one pushing you harder to meet any deadline.

More importantly, you can make your own questions and start investigating them. This way you will not have people behind you to change your project according to their requirements.

Being said all that,

Research can come from anyone and anywhere. The whole point of research is to add something to our collective knowledge.

If you think you can do it, you are more than welcome.

Hope this helps. Let me know if you need anything else.

Irfan