Archimedean Hecke Algebra of $\operatorname{GL}_1(F)$ for $F = \mathbb{R}$ or $\mathbb{C}$

Question

Suppose we have a $\operatorname{GL}_1$ over a number field $F$. I am interested in a description for the archimedean Hecke algebra (always taking the maximal compact subgroup). We know will be the tensor product of the archimedean Hecke algebras on the real places and the complex places. So that reduces the problem to a description of the Hecke algebras for $\mathbb{R}^{\times}$ and $\mathbb{C}^{\times}$. We have for $\mathbb{R}^{\times}$ that $K=\{1,-1\}$. That should mean that the Hecke algebra should be given by derivatives of all orders at $\{1,-1\}$, but I can not get a good sense of the convolution product.Is this isomorphic to something more familiar? The case for $\mathbb{C}^{\times}$ is even more confusing. A reference for the archimedean Hecke algebra will be also good. I only know the book Cohomological Induction and Unitary Representations by Knapp and Vogan, but there is no copy in my library.