Why can't quantum field theory be complex instead of imaginary?

In the following question 1, the author claims that a QFT is defined as:

$$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$

Then uses this definition to explore the possibility of formulating a QFT using the quaternions, on the grounds that it is constructed over the complex numbers thus why not try to extend it.

Is this definition of $Z$ correct? My understanding is that the correct definition is:

$$Z[J] \propto \int e^{i \left( S[\phi]+J.\phi \right)} D[\phi]$$

In this case, the exponentiated term $i(S[\phi]+J.\phi)$ is not a complex number, but only an imaginary part.

The original author asks: "Why can't quantum field theory be quaternion instead of complex?"

First, I would like to confirm if the author's definition is or isn't an error. Then, assuming that is it an error, I would like to ask the intermediary question: is there any possibility of a QFT which admit a real scalar within the exponential term in addition to the imaginary term, such that the sum is over the complex numbers and not just the imaginary part?