I'm among a limited but growing number of scientists who believes that believes consciousness remains active during dreamless sleep. The most obvious example of this is what you might call an "imageless lucid dream" which is where you're aware that you're dreaming until the dream 'collapses' and you remain conscious, just existing in a black void. You can see nothing but black, hear nothing except the internal vocalisations of your own thoughts, taste, smell and touch nothing, you might feel limited or no emotions - just peace (a burst of strong emotions at this point would tend to cause you to wake up).
If consciousness did exist after death, I imagine that is, at a minimum, what it would 'feel' like. But from a purely neurological point of view - consciousness ceases with death.
There are other angles you could attack the problem from such as the various arguments from quantum physics, but I never trained or studied those areas - so whilst I couldn't say it's impossible, I personally don't believe that consciousness continues post-mortem.