When our neurons die, we say the person has died by definition. However, nobody knows what really happens and if the person truly loses conscience. We barely understand how the brain even works and are a long way from being able to answer such a question.
Many people believe in a spirit that continues the conscience in some way, even after the neurons are gone. Our brains have a 100 billion neurons and trillions of connections, so all the information might just get transferred to something else we don't yet understand, preserving the conscience in a new way outside the brain.
