I have a reason to believe that Euler introduced the complex potential in his Continuation des recherches sur la theorie du mouvement does fluides, published in 1757. However, I am having hard time recognizing where it appears. The relevant place seems to be here:

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After establishing that $(u-iv)(dx+idy)$ and $(u+iv)(dx-idy)$ are integrable, he introduces functions $\phi$ and $\psi$. The problem is I am not sure what $\phi:(x+yi)$ means. Normally, I would expect the expression to involve some kind of derivative, such as $$ (u-iv)(dx+idy) = d\phi + id\psi. $$

I am hoping somebody could explain what is going on here.

If you need, Euler continues on the following page as:

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