Deep is an adjective, so can only be used to describe the quality of a noun:
How deep is the water?
Deeply is an adverb, so can only be used to modify a verb:
Were you thinking deeply?
Deep is often used instead of deeply, to the extent that it is largely accepted without comment. However, I've never heard it the other way around. "What a deeply question" is just plain wrong.
Being a possible collocation of appreciate, "deeply appreciated" would be more suitable. Dearly, on the other hand, will also intensify the meaning of appreciate by adding a sense of "very much".
Best Answer
In that context, it's deeply.
Deep is an adjective, so can only be used to describe the quality of a noun:
Deeply is an adverb, so can only be used to modify a verb:
Deep is often used instead of deeply, to the extent that it is largely accepted without comment. However, I've never heard it the other way around. "What a deeply question" is just plain wrong.