I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard examples thereof, are essentially, or are supposed to be, generalizations or algebraic versions of singular cohomology. If I am incorrect in this assessment, please do correct me.

Meanwhile, we have other interesting cohomology theories in topology: for example (topological) K-theory, elliptic cohomology, complex cobordism, .... Correspondingly, then, are there notions of "K-Weil cohomology theory" or "elliptic Weil cohomology theory", etc.? Is it possible?