Does exponential acceleration affect measurement of light speed?

An observer will measure a constant speed of light from a fixed light
source regardless of his velocity relative to the light source, but
does this hold true even if the observer is accelerating? How about if
even the acceleration is increasing, or even increasing exponentially?
Do the answers remain the same if instead the observer is fixed and the
light source is accelerating?

Also, if a light source is unidirectional, at rest relative to an
observer, positioned at a distance from the observer, and pointing at
the observer, then the observer can see the light, regardless of the
light source's (and thus also the observer's) velocity relative to
anything else. However, if both the light source and the observer are
accelerating at the same rate, in the same direction, with that
direction being perpendicular to the line between them, and the
direction in which the light source is pointing still coincides with
the line between the light source and the observer, then although
they're still at rest relative to each other and the light source is
still pointing at the observer, the observer can no longer see the
light, right?

Do all of these results hold if the observer and light source are
moving through water instead of through empty space? (Besides the fact
that the measured speed of light will be lower.)

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