Whether David Chalmers believes that consciousness doesn't come from the brain depends on what you mean by "comes from." He believes that physics can completely describe the physical world; he does not believe in souls and interactionist/Cartesian dualism. Instead, he suggests that information may have both physical and mental manifestations.

While I once found this line of reasoning convincing, I'm no longer so sure it matters. While consciousness itself may be non-physical, under the assumption that physical effects have only physical causes, we should be able to explain why we say we are conscious in physical terms. I'm not sure that an explanation of the latter will also explain the former, but in the end, I don't think we will care. I think we will find that the everyday concept of "consciousness" mixes together many different ideas, each of which are simpler to solve on their own than they are when we consider them together.

Roughly a century ago we went through a similar process with the concept of life. People wanted to know what makes a thing alive, and many intelligent scientists were drawn to the idea that there is something innate to a living organism that makes it alive. After decades of biological research, we're still not sure what makes an organism alive, in the sense that the line between living organisms and inert biological material is not well-defined, but we know how living organisms reproduce and grow, and that's really what matters.