What is the definition of "Complexity" in physics? Is it quantifiable?

Being somewhere in the wide and diffuse field of complex-systems science, I am not aware of any generally accepted definition of complexity, let alone one that yields a measurable quantity. It’s more an “I know it when I see it” thing, and in my experience, there is common agreement that the term cannot and does not need to be defined rigorously (for reasons that I elaborate later).

I have many times witnessed that scientists dealing with more complex systems expressing a tongue-in-cheek superiority over those dealing with less complex but still complex systems, saying that they were not really complex. In another example, the 90-page review paper The Structure and Function of Complex Networks applies the adjective complex to networks only a handful of times and not at all in a way that could serve as a definition.

If I were to define the field, I would say probably say something along the lines of:

Complex-systems science investigates phenomena that emerge from the (complex) interplay of perfectly or at least well understood components.

Thus, if you so wish, you can define a complex system as one that is principally capable of exhibiting such emergent phenomena. Of course, these definitions inevitably inherit vagueness from terms such as emergence, well understood, or principally capable. Moreover, once we understand a complex system, it becomes a well-understood component itself.

However, once we go down to specific research, these intricacies of the definition do not matter anymore: As long as your research gains insights on phenomena that are not yet understood, it yields new knowledge – whether the system exhibiting the phenomenon is complex or not does not really matter in this respect.

As others have said, complexity is still a notion that does not have one and only one definition. However, you did ask for a quantitative definition, so one example of complexity of a quantum field (or even a quantum lattice system) I've seen in the context of quantum gravity is:

"the minimum number of quantum gates needed to take you from the ground state of the system (or really any distinguished state) to the state in question."

This definition is due to Suskind (pg 7) to the best of my knowledge. It's not the only way to do it, but generally, quantum information theory is interested in complexity of a quantum system and its relationship to entanglement.

I agree with the previous statements, that no formally and broadly well accepted definition of complexity exists. Still, an intuitive idea of what is complexity is being developed, somehow. The problem is that the formalization of such idea changes, considerably, from field to field (one example being that of Suskind, above, which is rather vage since it relies on the also vage concept of "any distinguished state"---the case of the "ground state" as a distinguished state may be misleading since there are systems whose ground state is considered to be complex on its own and, therefore, the definition cannot be applied without getting into a contradiction). A vage description of the idea is as follows:

A complex system is one that cannot be modeled using a model (both, deterministic or stochastic) with a small number of adjustable parameters. In other words, the amount of information required to provide the "minimal" or more "succint" description of the system is "large".

To get a better idea about this, you can read the book of Murray Gell-Man (The Quark and the Jaguar), the works of James P. Crutchfield and the work of Jorma Rissanen about the Minimum Description Length (MDL) principle. Gell-Man elaborates more on the intuitive idea. Crutchfield and Rissanen elaborates more on how to formalize the idea, although with quite different approaches; the former more on "complex processes" and the later more on "complex structures/states". All of them give an idea on how to quantify how much complex a system is.