No, but 1 - 2i is equivalent to 2 + i, in that they both divide each other (multiply by i to go from left to right, or by -i to go from right to left), and similarly, 1 + 2i is equivalent to 2 - i (just the conjugates of before). When people talk about unique prime factorization in these contexts, they always mean up to such equivalence, which is to say, modulo multiplication by units.
OK, that’s sufficiently obvious that I really should have noticed it.
Actually, the easiest (but not the simplest since I would have to define several terms) way to state unique factorization in a ring R without zero divisors, whose whose subset R* of non-zero elements thereby forms a monoid, and whose group of units is U, is that R*/U is free monoid. The generators are the equivalence classes of) primes. You do that and 1 is not a prime. And 0 is ignored. Note that while generally cannot form the quotient of a monoid by a submonoid, you can when the submonoid is a group, which the units always are essentially by definition of unit.
From that point of view the question would never arise.
Free commutative monoid, rather.
Absolutely right, sorry I misspoke.