Yes the clock and me and everything that traveled with me would be displaced in time. Any clocks would have an easy means of interpreting the time difference but everything else experiences the same effect.
You keep saying how easy this is to do, but you won’t listen to what anyone else is saying and it’s getting frustrating.
You need to get back to basics. What exactly are you planning on measuring with your oscilloscope?
I am asking what devices could be constructed that would allow the time dilation to be measured as was done in the two clocks experiment. I proposed that oscilloscopes could be used because some objected that the time difference is small. But it well within the ability of oscilloscopes.
So given that 2 microseconds is not a barrier, how might it be done?
I think I see what you are asking for, but “time displacement” isn’t like “being out of phase”, it is just being 2 ms younger than you would be otherwise. So what you would need is a record of your moment of birth accurate to the millisecond and a device that can measure your biological age that is accurate to the millisecond, which you must agree doesn’t make sense.
Sure. But as has been explained - there is nothing in your body that carries information to that accuracy. If you carried a clock that could maintain that accuracy, you could measure against it. But you didn’t.
Your age isn’t any sort of intrinsic thing. We had a thread a while ago about the churn of atoms in your body. Most what constitutes you now is not even the same atoms as what took those flights. But this doesn’t even matter.
You talk of displaced in time. This is probably the core problem. You are not displaced in time. There isn’t an absolute time to be displaced in.
Imagine there are two roads you could take to travel somewhere. One is longer than the other. You and a friend travel, you take one road, your friend takes the other. When you both arrive, one of you is not displaced in space by the difference in distance travelled. i can’t take out a measuring tape and measure the distance offset between you. However, if both of you had taken pedometers with you, they would read different distances travelled.
Your flights took you on a slightly different path through spacetime than someone remaining in place on the earth. Once you are back home, this distance travelled through spacetime is very slightly different But it is no different than taking a different road. There is no residual displacement. You exist at exactly the same location in spacetime as everyone around you. Your path to get there was very very slightly different.
You are asking for an instrument than can tell which road you took to get somewhere on the basis that you should be distance displaced.
The two clocks in the 2 clock experiment probably still exist. The earth bound clock being accurate will be in sync with other earthly clocks. The space traveled clock will still register a different time it is displaced by time dilation. True it is younger than the earth bound clock, but that is a cultural concept. It is in a different position relative to earth time and being a clock has a manifestation which my great grand daughter can observe. So if time dilation is a property of objects then it can be observed in me if I know how to manifest it.
Here’s the thing–it isn’t.
Just for emphasis. It isn’t displaced. This isn’t what happens. It took a different path to get here. But it is here in every respect.
Quite real. GPS satellites have to take into account running slower due to their orbital speed and faster due to being further out of Earth’s gravitational well. The two don’t cancel each other out so,
Combined, these sources of time dilation cause the clocks on the satellites to gain 38.6 microseconds per day relative to the clocks on the ground. This is a difference of 4.465 parts in 1010.[13] Without correction, errors of roughly 11.4 km/day would accumulate in the position.[14]
If I take different roads there is residual displacement that can easily be measured. Just as there was measurable residual displacement between the two clocks. I assume they still retain that measurable displacement.
Would it be different if only the clock case had traveled? Would it not be younger because there was no manifestation of elapsed time that you could have observed. Is time dilation only a property of readable clocks?
How? You are both sitting down to breakfast the following morning. What can I measure to be able to tell who travelled further?
To give another example, you travel on a spaceship around the Earth at nearly the speed of light.
You and your granddaughter start your clocks before doing so. Let’s say your clock runs for a year. Hers runs for something much greater, say 10 years.
You come back to Earth and your clocks are different. You can see obvious differences in age.
But when you are speaking to your granddaughter, having returned, you and she are experiencing time in the same way again. You speak at a normal rate, for example.
At the reunion meeting, you start a second set of clocks. These run at the same rate. One second on a clock is one second on the other.
Do you get that even in this extreme scenario, if you had NOT started the first set of clocks, there is nothing you can do to retrospectively measure the fact that you experienced time differently?
The clock case would be younger but you could never tell.
It would look the same as any other clock case. There is no possible measurement one could take upon it - retrospectively - to prove it is younger than any other clock case.
The “measurable displacement” is that one shows that the time is 12:01:01.000 and the other one shows that the time is 12:01:01.002.
Again, to show that difference on youself, you would need two things:
A birth certificate that states you were born on January 20th 1953 at 12:01:01.000.
A chronolizer that measures you Innate Body Chronotons and sees that they show you to have been born on January 20th 1953 at 12:01:01.002.
Yeah, I dare you to try it 
This is the key - now that you are both sitting down to breakfast after taking different roads yesterday, you can’t tell who went further. The only way you can tell who travelled further is if for example, you took note of the odometer reading of each car before and after you travelled. Or weighted the amount of fuel in each car’s tank before and after the journey. (or had a precise chronometer, synchronized them first, then took them along for the trip to measure each person’s time dilation) Either way, the spherical eggs in a vaccum taste the same that morning for both of you.
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Oh, I didn’t get the part about they end up at the same point. In that case if you have all of the parameters they would have traveled different distances at different rates.
If time dilation is not a residual property then did the clocks eventually reset to be equal? What is the decay rate? Would returning space travelers observe an unraveling world churning back to corrected elapsed time plus starting time? If not the property is residual.
But, you presumed your conclusion in your premise. That they would be in the same place so there is no measurable difference. I can’t argue with that.
I recommend spending a few minutes playing that video game, if you have not tried it already. You sure do observe some weird things when you go fast, but those effects “churn back” when you slow down and disappear once you stop.
As for two physically separated clocks, they will read differently, even if you bring them back together, and therefore have to be synchronised; there are protocols for doing that.