By the work of Voevodsky, A^1-homotopy theory, or Ciniski-Déglise, Triangulated category of mixed motives, we have motivic cohomology of arbitrary schemes (maybe noetherian and finite dimensional, but thus is the case here) with integral coefficients. For regular schemes, motivic cohomology with Q-coefficients is isomorphic to the K-theoretic version of motivic cohomology you mentioned, see Cisinski-Déglise, Introduction, Thm 10, footnote 12. So you get independence of the base field. See also their computation in Example 11.2.3.

Presumably this independence can be checked directly with Voevodsky's definition but I haven't tried to do it.