There are a few interesting things here. For one thing, while imply has a fairly similar meaning to the closely related implicate, the connection is not so clear for the pairs comply::complicate, duplicit::duplicate, or reply::replicate. How did this come about? And more generally, what is this common ply root?
So I spent some time at dictionary.com. For most of these words, -ply comes from Latin plicāre meaning "to fold" (indeed, plywood is wood in which many layers have been "folded" on top of each other), or the related plectī, "to plait/braid/intertwine". Interestingly, dictionary.com gives an archaic usage of implicate to mean "to fold or twist together; intertwine; interlace".
However, there are a couple of surprises. Comply instead comes from Latin complēre, where -plē- means "fill"; we also see this in words like plenary and plenipotentiary. Thus to comply with a request is to "fulfill" it. So we can see why comply and complicate have such different meanings; their history is different. Comply does not really belong in the table above.
Another suprise is that duplicit doesn't actually seem to be a word. I will regard this as a failing of the language and not my knowledge of it: if complicity, complicit, and duplicity are all words then it is only right and proper to include duplicit.
In any case, what can we say about the literal meanings of these words in light of our new knowledge?
- A complicated situation is one in which many factors are braided together.
- An accomplice is complicit in a crime if he is "folded into" the plot.
- If we say that "action implies an actor" we are saying that action and actor are woven inextricably together.
- A duplicit man is a braid of two disparate strands (perhaps that's a stretch).
- To reply to a question is to "fold it back" towards the questioner.
It's harder to come up with such literal interpretations of replicate and apply, but we can incorporate the sense of "folding" into our understand of the verbs.
The investigation also turns up a bunch of words with similar derivations to those originally listed. The -plex in complex and duplex also comes from plectī. Perplex and complexion are in the family too. At this point we are reminded of flex and all its relatives; flex comes from Latin flectere, "to bend", which seems similar in both meaning and spelling to the words we have been discussing, so it may be that there is some relationship; I didn't look into this.
Here is another striking example of the misleading similarity between comply and complicated: the almost identical words accomplish and accomplish have very different origins: accomplish is related to comply, while accomplice belongs with complicated.
And finally, another amusing coincidence: ply in Latin means "to fold" in English. In the other direction, the unrelated English suffix -fold, as in twofold corresponds to the Latin suffix -plus, as in duplus ("double, twofold"), which is unrelated but looks very similar to the Latin -ply!