My impression is that current thinking is that connectome is enough. Other things I can think of the could matter:
- potentials between neurons
- neurotransmitter levels
- interactions with other parts of the body
But I think all of these are expected not to matter much because, for example, humans seem to come back the same as they were before after short periods of brain inactivity aside from damage from oxygen deprivation. Similarly neurotransmitter levels change the "mood" or "flavor" of a person, along with other chemicals that interact with neurotransmitters, but similarly these seem to be transient effects that change a person but are also easily tunable and so not necessarily a key part of what it means to be a particular person.
I think there is some case that interactions with parts of the body outside the brain might matter. For example, I find meditating is difficult when I am sick and have changes to my breathing. This suggests that something about this is interfering with the normal process that lets me get into a meditative state. But arguably this could easily be simulated and is not differentiating between humans, so may not and probably doesn't matter.
Thus to the best of my knowledge connectome is probably "enough" although more wouldn't hurt.