the best form of using deep in a comparative form is adding (er) at the end of it rather than using more. cause deep is one syllabic adjective for this reason we should use (er) at the end of it.for more than one syllabic adjectives we use (more).in this sentence (you must dig deeper) is the best .form of using deep to transfer the meaning.

You are correct. Technically, it should be “dig more deeply” as deeper is a comparative adjective not an adverb. However, people say “dig deeper” more commonly because it’s shorter and it is what is called an alliteration. Hence,it sounds more poetic but you are correct though in that it is incorrect grammatically.

Deeply is an adverb, and both examples work here.

Both would be grammatically correct, but the phrase to "dig deeper" is more commonly used. Both "more deeply" and "deeper" are comparative forms of deep.