How to make sense of $\int _{\gamma }f(z)\,\left|dz\right|$?

Question

Context

I understand from [1] that to define the contour integral, let $f:\mathbb {C} \to \mathbb {C} $ be a continuous function on the directed smooth curve $\gamma$. Then the integral along $\gamma$ is denoted $$\int _{\gamma }f(z)\,dz.$$

For a research project that I am working on in pseudo-Riemannian I would like to define a "path integral" with the norm of the differential. I mean like $$\int _{\gamma }f(z)\,\left|dz\right|.\tag{1}$$

My questions are open ended and I am generally soliciting not just answers to my questions but broader comments. Any citations to integrals with the norm of a differential (as in Eq. 1) would be welcome.

Questions

1. Is it nonsensical to define a "path integral" as in Eq. 1?

2. Can I restrict the functions $f$ so that the "path integral" in Eq. 1 makes sense?

3. Can I restrict the smooth curves $\gamma$ so that the "path integral" in Eq. 1 makes sense?

4. Can you provide any citations to a book that has an integral with the norm of a complex differential?

5. Are there any other points/details/objects that are important to consider as I attempt to work with "path integrals" as in Eq. 1?

Bibliography

[1] https://en.wikipedia.org/wiki/Contour_integration