Not significantly. From Wikipedia:
Liquid water can be assumed to be incompressible for most purposes: its compressibility ranges from 4.4 to 5.1×10−10 Pa−1 in ordinary conditions. Even in oceans at 4 km depth, where the pressure is 400 atm, water suffers only a 1.8% decrease in volume.
TL;DR: At 1000 m depth water is about 0.5 % denser and 1 % less viscous than at the surface, using the NIST chemistry webbook calculator.
First (this result surprised me a bit): How "thick" some fluid is, is something we measure as "viscosity". Fluids with high viscosity are "thick" (like honey) while for example air has very low viscosity. Viscosity is a function of temperature and pressure (or temperature and density, your choice). At surface conditions and 277 K (0.1 MPa and 4o C) water has a viscosity of about 1574.9 micro Pa s, while at 10 MPa and 4o C (1000 m depth) it has a viscosity of about 1559.8 micro Pa s, so about a 1 % decrease. Which means water actually gets less "thick" as you go deeper.
Second: The density of water (how "heavy" it is) varies slightly with pressure and temperature (as mentioned by @sylveon@lemmy.blahaj.zone). For example you might have heard that water is densest at 4o C (at atmospheric pressure). So at the surface, where the pressure is about 0.1 MPa water has a density of about 999.97 kg / m3, while at 1000 m depth, where the pressure is about 10 MPa it has a density of about 1004.9 kg / m3, about a 0.5 % increase. However, for most practical applications it's perfectly fine to assume that water is incompressible (as mentioned by many others).
Of course, both density and viscosity also depend on salt content, other impurities etc. but including that in computations is orders of magnitude more difficult than doing computations for pure water, and pure water is probably a good indicator. You can check out more for yourself using the NIST chemistry webbook calculator.
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It does vary sightly, but it's due to the temperature difference as you go down, rather than the pressure.
Water basically does not compress so no.
To answer very literally, no. The water itself isn't thicker, it's the same. It is denser because the pressure is significantly higher, because the water is colder, and to a small degree because salt settles down
However I also wanted to consider what's in the water so I did a bit of research and while there's a higher density of salt at the bottom of the ocean there's more bacteria and other small organisms higher up toward the top of the water. So if you're only thinking about pure water than it's the same but if you're trying to compare water content it seems to be thicker / more condensed higher up
It is, but I'm not smart enough to explain why.
Pressure
