According to Kevin Tobin Julius Caesar wanted to start the year on the vernal equinox or the winter solstice, but the Senate, which traditionally took office on January 1st, the start of the Roman civil calendar year, wanted to keep January 1st as the start of the year, and Caesar yielded in a political compromise.

However, besides keeping track of time through the phases of the Moon, one can also keep track of time by the path the Moon takes through the sky. Using the course of the Moon to keep track of time results in using what modern astronomers call a sideral month, which is 27 days 7 hours and 43 minutes long. Every 27 days the moon returns to the same position in the sky it was 27 days before. The scholar Vaster Guðmundsson believed that this was the form of month the Norse used, and used it in his theoretical reconstruction of the ancient Scandinavian calendar (Guðmundsson 1924, p.88). It is possible then that the Anglo-Saxons also did the same. However, as Bede draws a comparison to the Greek and Hebrew calendars, we may want to assume that the Anglo-Saxons used a synodic month (a month measured from a phase of the Moon to the next time that phase of the Moon occurs). There are other clues in Bede's account, that indicate this was so, and I will touch on those later. Bede then goes on to name the months of the old Anglo-Saxon calendar and further gives the corresponding Roman month.

So just treat years, months, and days as you would hours minutes and seconds in a 24-hour clock. I've also been doing some thinking about how to represent specific durations and placement like "month 2 day 3 of each year" could be written by appending an M to the designator meaning that the first unit represents months and the duration is equal to the last unit (in each unit equal to the one [that general precedes] the first unit, in this case 'each year'). So "month 2 day 3 of each year" becomes 2.3TCM. And as for [another example,] "day 0 of each month" becomes 0TCD

When did you turn 1 year old?

Quarters and Weeks within a terran computational date are not officially defined or supported, they are only accomodated through datemods. It's up to individuals/groups to create any further definitions of what a quarter or week may be for their own purposes.

This calendar does not track the moon, or all seasons, it only tracks days and the year (which always falls within a day or two of the n. winter solstice). As for quarters, these are useful for business and industry and it's pretty easy to remember their dates since they're all 3 months and 1 week:
44TC+0Q = 44.0.0 TC
44TC+1Q = 44.3.7 TC
44TC+2Q = 44.6.14 TC
44TC+3Q = 44.9.21 TC
44TC+4Q = 44.13.0 TC

+4Q probably shouldn't be used for businesses [but they should just instead] include 'minimonth' in the last quarter, but that's not in the scope of the terran computational algorithm.

We can already do that.

Which is why I raised doubts that a computer would be programmed to do it correctly to begin with (it will only be on the minds of programmers near that 128-year threshold), and doubts that it would be remembered at all ("Just let the machine do it").

Only if you ignore your epagomenal days. Food and fuel must still be consumed, rent must still be paid, and interest must still be accrued. Your perpetual quarters are actually 91.310 546 875 days exactly.

Of dubious value, as tropical seasons are not of equal (or uniform) length, and the beginning and ending dates of your quarters will be non-obvious.

But the relationship between cardinal and ordinal dates will become ambiguous.

Every single law and contract mentioning "the first of the month" will need to be rewritten to be unambiguous, since "1st day of the month" and "day 1" will be two different days.

There is no such thing as a perpetual calendar. Comparisons between algorithms of intercalary days can only be valid for about one millennium from now. Even the current standard of leap seconds itself (which allows for up to 12 adjustments per year) will break down around then.