Two real scalar fields $\phi_1$ and $\phi_2$ satisfying an $SO(2)$ symmetry and one complex scalar field $\psi$ are equivalent. However, the latter is more convenient because $\psi$ and $\psi^\dagger$ form the antiparticle pair, while in the real case, you need to change basis from $\phi_1$ and $\phi_2$ to $\phi_1 \pm i\phi_2$. Once you do this, you just get the exact same thing as the $\psi$ field.

This is explained really nicely starting from p.53 in [these notes](http://arxiv.org/pdf/1110.5013v5.pdf), which motivate the complex scalar field from scratch.

However, as you said, a complex quantity is not measurable in QM. And indeed, in QFT, $\psi$ is not an observable, which feels really strange because quantum fields are often motivated, at the very start of a QFT course, as nice local observables. This really confused me too. However, if you look at applications, you'll see that quantum fields never are directly observed -- for example, if you want to measure particle number, you would look at $\psi^\dagger \psi$ or $\phi^2$, etc., never the field itself. You only measure field bilinears.