Removing the continuity assumption, perhaps even time, kind resolves the duplicate question: both copies (if conscious) should have a memory of conscious you and experience their consciousness being a continuation of that, while you, now you, are as removed from those future copies, as you are from me.
One question, that seems tied to consciousness, is what is "now"? Are there objective reasons to believe in the existence of a canonical "now" existing independent of conscious experience? If not then there really is no point in distinguishing either momental or continuous experiences as special in any way. Perhaps consciousness just emerges as an interpretation of a single state connecting past and future by memory and anticipation, it's simply appears continuous like the series of images at the top of the article appears to be.
What I perceive as a continuous consciousness is really just a memory of other "I"s who were caused by this brain in the past, left a memory of introspection and then dissipated to be replaced by an new one.
This view has many advantages: it solves the problem of time you elude to, it answers the problem of other minds and the problem of my childhood self. The only problem I can not see solved is the hypothetical molecular level duplicate I mentioned above.
>So you have the question of why I can only directly perceive that emergent property in exactly one brain and not others
You are the thing carrying around that brain and sense organs, and the plumbing thereof. This is a unique situation, hence the uniqueness.
> Likewise you'd have to explain why the emergent property is apparently continuous through my whole lifetime over which my brain changes enormously.
Emergent phenomenon can be very robust, such as Jupiter's spot which was observed 340 years ago, or the common orbital plane of the solar system, which is billions of years old.
Also, your consciousness is not continuous. You turn (at least some of) it off when you go to sleep. Very curious! It's at least robustly bistable!
> Finally you'd have to deal with the hypothetical case where I am duplicated molecule for molecule and then ask which consciousness would I have direct access to, the duplicate or the original.
This seems very simple too. There are two copies of you at the instant of duplication, which diverge from that moment on. You1 would have access to the consciousness of You1's nervous system and You2 that created by You2's nervous system. This question is only difficult if you insist on something other than the bodies being involved.
I put you in a duplication chamber. In front of you is a card with the letter "A"
In a neighboring identical empty chamber is card with a "B" on it.
You close your eyes and I pull a big steam-punk lever and an exact copy of you is made in chamber B
When you open your eyes, what letter do you see and why?
But the original observer that was put into the duplication chamber, would maintain their letter A
To say somehow the different history of the atoms in the original brain vs the atoms in duplicate is somehow relevant to the emergent property would be a very odd claim: we don't take into account the prior history of the atoms in turbulent flow for example, just what the are doing right now.
Or for clarity, if I gradually replaced every molecule in your brain with another identical one, the emergent "you" would not be affected. So, if it weren't already obvious, it clearly can not be dependent on the particular atoms that make you up.
And by the way, if your brain at 10 years old and 90 years old, in spite of being very different structurally, still give rise to your same "you", the copy would not even have to be very good.
It is a fun problem I must say.
In terms of "me", there would be two individual instances of me that open their eyes and either see A or B. The "original" is determined by you the question-asker, and the fact that that "original" walked into chamber A. Because that "original" walked into chamber A, that "original" would see the letter A, whereas the clone would see the letter B. Its possible that the "original" was, at time of cloning, completely recreated/rearranged. However, that "original" would retain the A designation/"originality" because to an observer, someone walked into a chamber with no other exits, and exited out the same chamber, that someone is the same person.
You are as much what others think of you as what you think of yourself, in this case.
(Btw, the videogame _Soma_ goes over your exact scenario)
Put another way, using our current knowledge of physics and chemistry, we can, on a high level anyway, account for every item in the cause and effect chain from the photons hitting the retina to the arm that a person raises to shade his eyes. There is no room to insert anything into that causal chain let alone something so massively influential that it causes all the body actions that we consider driven by conscious choice. Thus whatever is going on must be made from very ordinary matter.
It's this very fact that makes it such a fun puzzle to look at, very ordinary mater maybe doing something (causing consciousness) that is wildly unexpected.
Unlike turbulence, I only have one case where I have ever seen this consciousness which I am trying to explain as an epiphenomenon, namely my own. All others consciousness (if they even exist which I can't prove directly) are not remotely observed in the same way, they are self reported by other people.
So you have the question of why I can only directly perceive that emergent property in exactly one brain and not others.
Likewise you'd have to explain why the emergent property is apparently continuous through my whole lifetime over which my brain changes enormously.
Finally you'd have to deal with the hypothetical case where I am duplicated molecule for molecule and then ask which consciousness would I have direct access to, the duplicate or the original.
So while dualism is obviously patently absurd, the idea of consciousness as epiphenomenon, while promising, needs work too.