Is "very much" an adverb or an adjective?

Classically, an adverb can modify either a verb or an adjective (or perhaps other things), but there are subtleties. "Very" is an adverb that can modify adjectives but not verbs, which is useful to know for a grammarian, because it can be used to distinguish verbs from adjectives in cases where the difference is not obvious (such as participles).

At least sometimes, when "very" cannot be used because it would modify a verb, it can be converted to a verb modifier by adding "... much":

The new regulations bewildered him.  
He was very bewildered at the new regulations.  
*The new regulations very bewildered him.
?? He was very bewildered by the new regulations.  
The new regulations very much bewildered him.  
He was very much bewildered by the new regulations.