For my generation, the normal response to How are you? is I’m very well. Here, well is a predicative adjective, the sentence being on the pattern Subject – Verb – Complement and not, as some might think, Subject – Verb – Adverbial. A younger generation, to the indignation of the huffers and puffers, now tends to say instead I’m good. Here, too, good is an adjective and the sentence is on the same pattern. There are no grounds for objecting that good might imply some moral quality. It is quite clear from the context that what it means is something like in a healthy state of mind and body. Just like well, in fact.
We can also say I feel well and I feel good, but here there is a difference in meaning. The first suggests the speaker feels healthy, the second that the speaker has a less specific sense of well-being. While well is perhaps most frequently found as the adverbial equivalent of good, the use of good itself as an adverb is non-standard, as in The boy done good.
It’s a little different when things are not so good. I’m bad can mean that my behaviour in some way falls short of what society requires and in some contexts it means I am sick. What happens when we change the verb and say I feel bad? That, too, can mean that I’m feeling unwell, but when we have done something that might have upset someone, we might say I feel bad about that. Is that Standard English, or does Standard English require I feel badly about that? Those who would condemn such usage make the same mistake as those who object to I’m good. They see bad used wrongly as an adverb modifying feel. Seen as its complement, however, it’s just as much Standard English as I’m good. By contrast, We didn’t do too bad is non-standard.
The point may be clearer in another example. A recent BBC publication has the title ‘Cook Healthy’, which I take to mean ‘cook in a way that promotes health’, ‘healthy’ being the complement of ‘cook’. It’s different from ‘cook healthily’, which would mean, if it meant anything at all, ‘cook in a way that promotes your own health while you’re doing it: make sure you have plenty of fresh air in the kitchen, and take frequent breaks from the stove’.