For the purposes of this thought experiment, let's assume that consciousness is an emergent property of our brains and sensory organs. When you plus our senses into a processor, consciousness is created. This is broken down into the formula : inputs + processing = consciousness (i + p = c)
Now, if you were to create a clone of yourself with exactly the same memories and physical properties as you in that moment in time, then your consciousness should also be duplicated. However, it stands to reason that you and the clone would not have a shared consciousness. The obvious reason why is that the formula has been broken. Despite the clone sharing your sensory organs, memories and brain chemistry, the real world input is different. "i" for the clone is different, causing "c" to change too. As the clone stands looking at you, all of its sensory inputs are mismatched to yours. But, what if the sensory inputs were matched? If the clone emerged in exactly the same space and time as you, would you both share a consciousness?
The end result of this may seem pointless as no matter where or when the clone is positioned, it's sensory inputs will never match the original you. Unless the clone exists in a parallel universe. If the clone and you exist in two separate but identical universes, only then will the inputs and processing both match. Meaning that if this equation is assumed to be true, the consciousness of the two entities would be shared. Of course if the parallel universe altered from the original, then it's fair to say that the shared consciousness will split as well.
If parallel universes do exist, this logic asserts that a shared consciousness between universes is at least logically
Possible