Time lag between observed electrical field and polarity of light?

I was listening to the Feynman lectures on physics Vol. 3, Ch.2 the other day, and he was talking about the relationship between light polarity and the electric field of light as it is transmitting through dense material.

Suppose a medium is uniform and isotropic. Suppose light passes through the medium. Light is an electro-magnetic process that interacts with the atoms of the medium. The atoms in the medium all have electric dipoles.

The dipole moment is proportional to the ambient electric field.

When an electric field is externally provided, there exists an overall dipole moment in the material (my interpretation: by and large, the dipoles of the atoms in the material are lining up in the direction of the electric field). However, the dipole moments are still noisy — the electric field is effectively a guideline, rather than a rule.

As light passes through the material, the individual photons are gradually torqued into a polarity that lines up with the over-all dipole moment of the material, and this torque corresponds to a classical analysis with Maxwell’s equations.

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He was saying something to the effect that: as light passes through a uniform and isotropic polarizing medium in the presence of an external electric field, the polarity of light lags slightly behind the polarity of the electrical field due to complex effects.

Specifically, he says:

"There is a complex part [as in complex component of the wave] means that the p — the polarization — does not exactly follow the electric field but may delay behind it a little bit. At any rate, there is a polarization per unit volume proportional to the electric field."

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I can think of why, in real statistical terms, the polarization properties would lag a change in the electric field.

1. the atoms’ dipoles are unstable and noisy, so perfect alignment cannot occur instantaneously due to signal noise received by the photons — the local neighborhood of atoms "vote" which way the photon should lean, and so its polarization is the evolution through time of a massless optical billiard ball with an electric field that snaps to-and-fro depending on the neighborhood it is on. — seems like the lag here would be gyroscopic…and without mass, virtually instant.
2. statistically, the light has to be on the high side of its EM-oscillation when it is near an atom to be affected — perhaps this is the source of the "complex" effect? — explains the lag en-masse, but an individual photon could be tuned to exhibit instant changes.
3. the non-linear components of light-matter interactions are by and large neglected, and probably account for any un-explained behavior on the margins. — I don’t know much about these processes, but they seem to be more of an issue at the interface of a wave-guide and an opaque housing rather than the transmission through the transparent wave-guide itself.

But, instead, the statement is that "there is a complex part" (i.e., a component in the imaginary dimensions), and the complex component causes the lag of the polarization behind the electric field.

How does this work? To my mind, the "complex" part of "complex analysis" is an axiomatic trick invented by Euler to make math more workable by hand. There is no reason at all for there to be conformity between physical processes and imaginary numbers. The primary interesting property of complex numbers, relative to real numbers, is collectively the nth roots of unity. In all other regards, complex numbers are equivalent to real numbers. Note: I am not marginalizing complex analysis here; rich and dynamic processes can be introduced with this feature.

A property corresponding to an nth root of unity would have a quantized and wave-like nature with all sorts of proofs and computational toolkits to work with (i.e., the discrete Fourier transform). And I imagine there are other ways to leverage the machinery of complex analysis, such as relating exponentiation of a phase to a unit of time (one exponentiation per unit time step), or using it as some pseudo-discrete metric for the decoherence process.

If I keep working this, a double slit (or "minimum verifiable diffraction grating") will produce something that looks exactly like the FFT of a binary (black and white) image of a square.

So, when Feynman says "there is a complex part" my antennae twitch.

Anyhow, these math techniques are ultimately analogies and models applied to real, physical processes. But, there do seem to be physical process that line up with the logic of an "nth root of unity" (even if the numerical axiom is imaginary and philosophical).

A good example is a solution of materials in a substrate that cause the color of the substrate to, say, change from clear transparent to an opaque blue after exactly n swirls of the flask (where n is determined by the mixture). But, although this process conforms well to the logic of a "complex" nth-root of unity, the physical concept is not mysterious.

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So what is the "complex part", in physical terms, and what is it doing that causes the time-lag between the light polarity in the medium and the electrical field?

Is it some sort of permittivity/permeability transmission constraint that throttles how fast the light can be "informed" of a new state it can occupy?

Maybe even a phased-array style diffraction process w.r.t. the overlapping dipoles and light that somehow "becomes" physical reality after some period of time? (I would equate this one to mutating a sound passing through a lattice of speakers). If this is the case, it would be interesting (since, supposedly, everything diffracts).

Or is it merely that light can only be affected by standard materials when it is on the high side of its oscillation, and since the alignment of this high side and the atoms in the medium will be random, there is a statistical expectation of lag that does not actually have to occur if it is perfectly controlled for?