Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can study also the $K$ theory of $\mathbb{C}$ with discrete topology (with possibly some minor modifications to deal with uncountable-dimensionality). Weibel, in his $K$-theory book, computes the torsion in its coefficient ring. However I can't find a computation of the full coefficient ring anywhere. The best language for this might be in terms of motives (without factoring out $\mathbb{A}^1$), but I don't know where to find its homotopy groups computed in this language either. Anyone know this?

Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete topology. Weibel, in his $K$-theory book, computes the torsion in its coefficient ring. I would like to know the torsion-free part in the homotopy groups, but can't find this anywhere. The best language for this might be in terms of motives (without factoring out $\mathbb{A}^1$), but I don't know where to find its homotopy groups computed in this language either. Anyone know this?