Interesting...

in base ten
144 x 441 = 252 x 252

and in base twelve
*144 x *144 = *252 x*252

and in base 6
144 x 144 = 252 x 252


...

In base b, "144" = b² + 4b + 4 and "441" = 4b² + 4b + 1
So ,144 × 441 = (b² + 4b + 4)(4b² + 4b + 1)
= 4b⁴ + 20b³ + 33b² + 20b + 4

Similarly, 252²
= (2b² + 5b + 2)²
= 4b⁴ + 20b³ + 33b² + 20b + 4

Therefore, it works in every base.

Furthermore, in bases 34 and higher, the product is 4KXK4. Technically, that works in any base if you don't restrict digits to being less than the base.

Here's another decimal one:
2 x 819 = 9 x 182
and the dozenal:
2 x T1E = E x 1T2