multiple definition in header file

The problem is that the following piece of code is a definition, not a declaration:

std::ostream& operator<<(std::ostream& o, const Complex& Cplx) {
   return o << Cplx.m_Real << " i" << Cplx.m_Imaginary;
}

You can either mark the function above and make it "inline" so that multiple translation units may define it:

inline std::ostream& operator<<(std::ostream& o, const Complex& Cplx) {
   return o << Cplx.m_Real << " i" << Cplx.m_Imaginary;
}

Or you can simply move the original definition of the function to the "complex.cpp" source file.

The compiler does not complain about "real()" because it is implicitly inlined (any member function whose body is given in the class declaration is interpreted as if it had been declared "inline"). The preprocessor guards prevent your header from being included more than once from a single translation unit ("*.cpp" source file"). However, both translation units see the same header file. Basically, the compiler compiles "main.cpp" to "main.o" (including any definitions given in the headers included by "main.cpp"), and the compiler separately compiles "complex.cpp" to "complex.o" (including any definitions given in the headers included by "complex.cpp"). Then the linker merges "main.o" and "complex.o" into a single binary file; it is at this point that the linker finds two definitions for a function of the same name. It is also at this point that the linker attempts to resolve external references (e.g. "main.o" refers to "Complex::Complex" but does not have a definition for that function... the linker locates the definition from "complex.o", and resolves that reference).

Move implementation to complex.cpp

Right now after including this file implementation is being compiled to every file. Later during linking there's a obvious conflict because of duplicate implementations.

::real() is not reported because it's inline implicitly (implementation inside class definition)

I was having this problem, even after my source and header file were correct.

It turned out Eclipse was using stale artifacts from a previous (failed) build.

To fix, use Project > Clean then rebuild.