Does this modified random walk (2D) return with probability 1?
This is an example of a persistent random walk (these are also called Newton's random walks). Its recurrence is a consequence of a general theorem due to Jean-Pierre Conze on the recurrence of so-called dynamical cocycles.
For an introduction to persistent random walks and a sketch of the application of Conze's result to the specific case you asked about, see Theorem 2.5 of this paper and the references therein.