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.