in 1983, makar-limanov showed that the quotient division algebra of the complex weyl algebra contains a copy of the free algebra on two generators. this results shows that, unlike in the commutative case, noncommutative localization can behave very pathologically. stafford and makar-limanov conjectured that the following general dichotomy should hold: if a division ring is not finite-dimensional over its center (essentially commutative) then it must contain a free algebra on two generators. we show that for division algebras with uncountable centers a weaker dichotomy holds: such a division ring must either contain a free algebra on two generators or it must be in some sense algebraic over certain division subalgebras. we use this to show that if $a$ is a finitely generated complex domain of gelfand-kirillov dimension two then the conjectured dichotomy of stafford and makar-limanov holds for the quotient division ring of $a$; that is, it is either finite-dimensional over its center or it contains a free algebra on two generators. this is joint work with dan rogalski.