What is the purpose of washing 2-chloro-2-methylbutane with saturated sodium bicarbonate?

I think it depends on the circumstances. In my lab we used NaHCO3 to wash a mixture of n-butyl alcohol and n-butyl bromide. The NaHCO3 washed out the unwanted n-butyl alcohol in order to purify the n-butyl bromide component.

In addition, the salt could be used to neutralize your organic layer.