In severe cases of epilepsy, doctors have sometimes performed a surgery called a callosotomy. It severs the corpus callosum, the nerve bundle in the brain that allows the left hemisphere to communicate with the right. The idea was that if a seizure then occurred, it could not spread throughout the whole brain. The surgery has been quite effective in most cases.
But what are the consequences of walking around with a split brain? A neuroscientist, who later won the Nobel Prize for his work, set out to investigate that question. The results of his case studies taught us a lot about hemispheric specialization, but also led him and other neuroscientists to arrive at a pretty audacious hypothesis: that the left and right hemispheres of a split-brain patient are independently conscious.
... indeed a conscious system in its own right, perceiving, thinking, remembering, reasoning, willing, and emoting, all at a characteristically human level, and ... both the left and the right hemisphere may be conscious simultaneously in different, even in mutually conflicting, mental experiences that run along in parallel— Roger Wolcott Sperry, 1974
So the question: Are they separately conscious? Is that a reasonable theory to draw from Sperry's findings and the full case history?
This will inevitably come down to what the definition of "consciousness" is. And there is no agreed upon definition of consciousness. Therein lies the difficulty. It's a question that isn't even sure what it's asking. A recent study declared split-brain patients have "divided perception but undivided consciousness". Yet where does perception end and consciousness begin? And how is the brain able to create a unified consciousness after it has been completely severed in two?