We introduce a new so-called distance complexity measure for Turing machine computations which is sensitive to long-distance transfers of information on the worktape. An important special case of this measure can be interpreted as a kind of buffering complexity which counts the number of necessary block uploads into a virtual buffer on top of the worktape. Thus, the distance measure can be used for investigating the buffering aspects of Turing computations. In this paper, we start this study by proving a tight separation and hierarchy result. In particular, we show that a very small increase in the distance complexity bound (roughly from
+ 1) +
) brings provably more computational power to both deterministic and nondeterministic Turing machines. For this purpose, we formulate a very general diagonalization method for Blum-like complexity measures. We also obtain a hierarchy of the distance complexity classes.