If you define a complex number as an ordered pair $(x,y)$ of real numbers, then the set of real numbers is not a subset of the set of complex numbers.
That is a problem that is easy to fix. The complex numbers of the form $(x,0)$ are a "copy" of the reals in the complex numbers. We informally identify this copy of the reals with the real reals, and then don't worry about it. The way that the addition and multiplication are defined on the complex numbers makes the arithmetic of the ordered pairs $(x,0)$ the same as the arithmetic of the reals.
In technical language, there is a natural isomorphism between the reals and their arithmetic and the complex numbers of the shape $(x,0)$, with the arithmetic inherited from the definition of sum and product of complex numbers.
By the way, the sum of two complex numbers $(u,v)$ and $(x,y)$ is defined to be $(u+x, v+y)$. The product of these two complex numbers is defined to be $(ux-vy, uy+vx)$. Calculate $(0,1)$ times $(0,1)$. You will find something interesting!