If two clocks are at different heights, they can compute the differential in their rates and agree on a time unit. Sure, it might be 1 quadrillion ticks in on one clock and 1 quadrillion and 1 ticks on the other, but it's still a 'second' so long as both sides see it as simultaneous.
With all this uncertainty, all you could do, after getting two clocks to approximately the same altitude, would be to say that clock A is currently running a factor of X faster than clock B. What you wouldn't know is how much of the delta was due to altitude, how much due to unequal density underneath, and other such factors, compared to how much was due to the internal differences in the clocks themselves that you were trying to factor out by using multiple clocks.
As a geologist, I really wish that were possible, but what you're describing is just far too small of an effect to be detectable from the surface. Gravitational acceleration follows an inverse square law. Any signal from flow in the outer core is completely swamped by the far larger (at the surface) changes from things like local hydrological conditions, ongoing human activity, etc.
On the bright side, though, we can already measure changes in the acceleration of gravity incredibly precisely without directly using relativistic effects. Even from orbit, things like regional flooding and changes in the mass of ice sheets can be detected purely from changes in acceleration of gravity. Have a look at some of the work that came out of the GRACE mission (also see the ESA's GOCE satellite): http://www.jpl.nasa.gov/news/news.php?release=2004-224
I'm having a lot of fun being wrong today!
EDIT: Aaaand that's from 2004. Which means I really wasn't paying attention while in college. Considering my roommate went to JPL to work on a sister project, I feel a bit of shame just learning about it now.
So even if this difference is undetectable for geological purposes it could very well screw up really accurate time measurements. It all depends on whether or not those effects compound or cancel out, in the case of gravity they will likely only compound.
It's not that timekeeping gets meaningless- it just means that there is a threshold where the variability of unsynchronized clocks disrupts certain highly precise problems.
More pragmatically, we'll probably agree on a standard centralized time (i.e. International Atomic Time) and everyone will translate between their own locally perturbed time and the standardized time when synchronizing.
In a sense, this is not a property of the clocks, but rather just the way nature is. We can't keep these clocks in sync at various places around the world, because the thing they measure is not in sync at various places around the world. But, we've known that for about one hundred years - what's the big deal?
nullc is right, this is a bad article.
This seems a bit sensationalist. The new clocks will be more accurate than any we have now, for existing applications. Those who develop new applications that actually use the increased precision will just need to be more careful about which frames of reference they use. Those who built GPS already solved some version of these problems.