Can a semicolon be used between two independent clauses that contain a question? For example, "Should I go at morning, afternoon, evening; or does it really matter?" Is that correct?
Yes it can but please read below for full discussion.
Before I dive in on this, please let me encourage you to read from the link below. It gets into way more detail and examples and things than I will here.
A semicolon is used to link two independent clauses/sentences. It acts sort of like a period in that it shows a break between the two, but it eliminates the breathing space associated with the period.
In the question, the sentence in question is:
Should I go at morning, afternoon, evening; or does it really matter?
As the sentence stands, the two clauses are not independent (the "or" makes the last bit a part of the choices of when the person should go) and so should be written:
Should I go at morning, afternoon, evening, or does it really matter?
If you want to have the "does it really matter" option to be equal to the option of actually going out, we could write it like this (with a period/question mark):
Should I go at morning, afternoon, evening? Does it really matter?
Or to eliminate the pause between the two clauses/sentences, we can use a semicolon:
Should I go at morning, afternoon, evening; does it really matter?
To throw my two cents into this (and without knowing any of the context for the thought being expressed within the larger work), might I suggest not using a list of when to go out and instead write as a musing, an internal dialogue, of someone , perhaps this way:
Should I go at morning? Afternoon? Evening? Does it really matter?
Which ever way you choose to go, always remember that the bottom line of writing is to express and communicate - grammar gives us rules to help with that expression but in the end it's all about having the reader experience what it is you want them to. If a semicolon gets you there, then use it.