Over Gaussian integers, can every prime be $p \equiv 1 \text{ mod }4$ be expressed as $(a + bi)(a - bi)$ in almost unique manner? Over the Gaussian integers, can every prime $p \equiv 1 \text{ mod }4$ be expressed as $(a + bi)(a - bi)$ in an almost unique manner, unique up to negation of $a$ and/or $b$?

Yes. The domain of Gaussian integers, $\mathbb{Z}[i]$, is a unique factorization domain, which means that each nonzero, non-unit number in that domain can be factored into primes uniquely without regard for order or multiplication by units. The units of $\mathbb{Z}[i]$ are $i$, 1, $-i$, $-1$.
Fermat (or was it Euler?) proved that if $p = 2$ or $p \equiv 1 \pmod 4$ is a positive real prime number, then $p = a^2 + b^2$, where $a$ and $b$ are purely real integers. You have noticed the connection to the norm function, which means that these numbers that are prime in $\mathbb{Z}$ are composite in $\mathbb{Z}[i]$, since then $p = (a - bi)(a + bi)$.
But then $(b - ai)(b + ai) = p$ seems to be another factorization. But note that $(b - ai)i = a + bi$ and $i$ is a unit of this domain. Likewise, $(b + ai)(-i) = a - bi$. What seems to be another factorization is in fact the same factorization, just with different units.
So for example, $5 = (2 - i)(2 + i) = (1 - 2i)(1 + 2i)$. This does not represent two distinct factorizations because $(1 - 2i)i = 2 + i$. It's the same principle why $21 = 3 \times 7 = -3 \times -7$ does not represent two distinct factorizations either.
