Questions & Answers about Я хочу побыть один.
Побыть is the perfective form related to быть.
- быть = to be
- побыть = to be / stay for a while, for some limited time
So Я хочу побыть один does not just mean I want to be alone in a general sense. It suggests:
- I want to be alone for a bit
- I want some time alone
- I want to spend a little time alone
That small difference is very natural in Russian.
In побыть, the prefix по- often gives the idea of doing something for a short time or for a while.
Compare:
- быть одному = to be alone
- побыть одному = to be alone for a while
This use of по- is very common with verbs when the speaker wants to express a limited duration:
- посидеть = sit for a while
- подождать = wait for a bit
- поговорить = talk for a while
So побыть sounds more natural than быть in a sentence about a temporary wish.