When you travel by boat from Southampton to Auckland, during the travel, you mostly only care about the flow of time on the boat, where it is, regardless of the time in the departure & arrival places. You only expect to be in Auckland, around a certain local time.
Going from Earth to Mars, or from the solar system to an Aldebaran planet is slightly the same (but that the flow of time itself may be relative between the places you left/arrived at, and the place you are):
- you expect to be somewhere in the future in about a relative time to where you are (I'll be there in what I know/can represent myself to be X-earth-years);
- you will account for inevitable delays during travel, that might be significant at your scale, but not to others';
- when arrived there, you'll try adjust to the local time keeping method that makes sense to locals - if only because that's how you will interact with them about the flow of time.
You may get to compute what's the relative time on Earth, at the same instant, or when they will get your message. Just as everyone do today (and before) when communicating through slow (aka, human-borne) package & mail delivery.
We can consider having an atomic (or similar) clock that pulse at the same rate, on both Aldebaran & Earth. But:
1. would they be still in sync? experience seems to say no; do we know how we would be even in a position to check that remotely?
2. would it be relevant, relative to Aldebaran-time-keeping? (the atomic second as it is used on Earth is 1/86400th of a full Earth day; it may be wildly on a different planet)
"See you in a kilosec!"
"Oh she got promoted a couple megasecs ago, transferred out."
"I've been up for a hectasec; I have to catch some Z's!"\
"I graduated early, only 5 gigasecs old!"