This factoid gets trotted about without ever noting the (tiny) magnitude of the effect, it just plants a (false) idea in the readers mind that you can get ice faster by microwaving water first.
> I mean the deeper you go the more freezes on it and the thicker it gets?
As the article makes a point of mentioning, unlike nearly all materials, added pressure is more likely to melt ice than cause it to freeze, at least at the pressures and temperatures we're considering.
The pressure of water at 10000m (basically the deepest the oceans go) is ~100 MPa. Look at the liquid part of the phase diagram between 100 kPa and 100 MPa. See how it abnormally extends left as pressure increases to ~200 MPa?
The only problem with that is you can't think of cold as something you can generate. Cold is the absence of heat, and so the only way to make it in a place where there is already heat is to move the heat somewhere else.
If you keep it inside the vessel, then you're faced with containing a zone of steadily increasing heat (and thus steadily decreasing efficiency of your heat pump). If you try to dissipate it outside of the vessel, you run into further engineering problems (the heat conduit has to pass through the hull at some point, which creates a weak spot for pressure and a potential heat leak to the rest of your hull).
http://en.wikipedia.org/wiki/Cat's_Cradle
http://en.wikipedia.org/wiki/Ice_nine (fictional material within Cat's Cradle)