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, 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.

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 the particles made by $\psi$ and $\psi^\dagger$ are each others' antiparticles. In the real case, the fields that have this property are $\phi_1 \pm i \phi_2$, so once you change basis from $\phi_1$ and $\phi_2$ to $\phi_1 \pm i \phi_2$ you've reinvented the complex scalar field.

This is explained nicely starting from p.53 in Sidney Coleman's QFT notes.

As you said, a complex quantity is not measurable in QM. And indeed  $\psi$ is not an observable, which feels strange because quantum fields are often motivated at the start of a QFT course as local observables. Unfortunately this motivation isn't quite right, as we rarely measure quantum fields directly. For example, the number density, charge density, and current density for a charged complex scalar field are all field bilinears like $\psi^\dagger \psi$, and hence real.

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, 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 a bit closer, you'll see that the quantum fields never are directly observed -- for example, if you want to look at particle number, you would look at $\psi^\dagger \psi$ or $\phi^2$, etc., never the field itself. So this isn't actually a problem.

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, 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.