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? 



What is the coefficient ring of algebraic K theory of the discrete $\mathbb{C}$?
I think the answer to this question is not known. All we can say about the K-theory of $\mathbb{C}$ concerns the torsion. 
The trouble starts with $K\_1(\mathbb{C})\cong\mathbb{C}^\times$, which is pretty difficult to understand as an abelian group. There is a formula for $K\_2$ of a field due to Matsumoto which is $K\_2(F)\cong (F^\times\otimes F^\times)/(x\otimes(1-x)|x\in F^\times\setminus\{1\})$ which you probably found in Weibel's book already. For $K\_3$ we still have a computation due to Suslin, which expresses $K\_3(\mathbb{C})$ in terms of a Bloch group, this can be found in the book "Homology of linear groups" by K.Knudson but probably also in Weibel's K-book. This description relates $K\_3(\mathbb{C})$ to scissors congruences in hyperbolic space and the three-sphere. However, although there is such a conceptual interpretation of $K\_3$, it is still far from understood: it is (I believe) still an open question if the natural inclusion $K\_3(\overline{\mathbb{Q}})\to K\_3(\mathbb{C})$ is surjective, this is a rigidity question attributed to Sah in Dupont's book on scissors congruences. Note that $K\_3(\overline{\mathbb{Q}})$ is "understood" in terms of Borel regulators.
Finally, there is a (still conjectural) generalization of the Bloch-group description of $K\_3$ to higher K-groups, due to Goncharov. You can find this for instance in Goncharov's article in the second part of Motives (Proceedings of Symposia in Pure Mathematics, Volume 55, eds Jannsen, Kleiman, Serre, 1994), or other articles of Goncharov.


