I'm looking for a fluid that is transparent, nearly as dense as water,
with low compressability, and very low viscosity (approaching that of
air at 1 atm), and preferrably non-toxic, all at standard conditions.
I'm just daydreaming but seeking details... any takers?

Your first 3 criteria (transparent, about as dense as water, low
compressability) are met by a variety of liquids (how many depends on
exactly what your criteria are, for example, how close to the density
of water do you need?).  The killer is the viscosity.  To the best of
my knowledge, *any* liquid will have a viscosity much higher than air.

If you would be satified by a liquid with a viscosity that was
significantly lower than the viscosity of water, you could probably
find some candidates.  But if you really need a viscosity "approaching
that of air" you're probably out of luck.  :(

Well, basically what I'm looking for is fluid (possibly even a very
dense gas?) that would be relatively easy for a suited diver to be
submerged in and still reach for, manipulate, and feel precision
controls for maneuvering a submersible vehicle effectively. I was
unsatisfied with water because of the fact that operator speed and
reaction time would be excessively slowed by the slow flow of the
surrounding liquid, but if you have a candidate that roughly (density
less than water but by no more than about 10%, compressability similar
to water [within a few percent]) meets the other criteria but and has
as low of a viscosity as you think I'm going to find I may be able to
compensate by specially adapting the operator controls for a thicker
fluid environment.