Your coworker is wrong, the common way is and always has been to put code in .cpp files (or whatever extension you like) and declarations in headers.
There is occasionally some merit to putting code in the header, this can allow more clever inlining by the compiler. But at the same time, it can destroy your compile times since all code has to be processed every time it is included by the compiler.
Finally, it is often annoying to have circular object relationships (sometimes desired) when all the code is the headers.
Bottom line, you were right, he is wrong.
EDIT: I have been thinking about your question. There is one case where what he says is true. templates. Many newer “modern” libraries such as boost make heavy use of templates and often are “header only.” However, this should only be done when dealing with templates as it is the only way to do it when dealing with them.
EDIT: Some people would like a little more clarification, here’s some thoughts on the downsides to writing “header only” code:
If you search around, you will see quite a lot of people trying to find a way to reduce compile times when dealing with boost. For example: How to reduce compilation times with Boost Asio, which is seeing a 14s compile of a single 1K file with boost included. 14s may not seem to be “exploding”, but it is certainly a lot longer than typical and can add up quite quickly when dealing with a large project. Header only libraries do affect compile times in a quite measurable way. We just tolerate it because boost is so useful.
Additionally, there are many things which cannot be done in headers only (even boost has libraries you need to link to for certain parts such as threads, filesystem, etc). A Primary example is that you cannot have simple global objects in header only libs (unless you resort to the abomination that is a singleton) as you will run into multiple definition errors. NOTE: C++17’s inline variables will make this particular example doable in the future.
As a final point, when using boost as an example of header only code, a huge detail often gets missed.
Boost is library, not user level code. so it doesn’t change that often. In user code, if you put everything in headers, every little change will cause you to have to recompile the entire project. That’s a monumental waste of time (and is not the case for libraries that don’t change from compile to compile). When you split things between header/source and better yet, use forward declarations to reduce includes, you can save hours of recompiling when added up across a day.