Comedy, today, is dead; murdered by political correctness.
The commas surrounding "today" are correct. You can also move the "today" to the front of the sentence if comedy specifically died today. However, the semicolon is incorrect. As you wrote, "murdered by political correctness" isn't an independent clause (in fact, it ...
According to The Cambridge Grammar of the English Language p1309,
Nor appears as a coordinator paired correlatively with neither,
or non-correlatively as a variant of or in negative contexts.
As this is clearly a negative context, the use of nor is prefectly acceptable.
Examples of this usage may be found in quality writing:
Ultimately, I saw a system ...
I prefer your version of the sentence: Comedy today is dead, murdered by political correctness.
If, however, a person chose to pause after comedy and is, that would be quite OK, as in the following sentence:
Mr. Kitchener, presumably, was responsible for the mishap. [Granted, the construction and parsing of this sentence are a bit different from your ...