  We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function
$d(n)$, that is, for a measure preserving $\mathbb Z[i]$ action $T$, the limit
  $$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N}
d(n) \,f(T^n x) $$ converges for every $f\in L^p$, where $\mathscr{N} (n) = n
\bar{n}$, and $D(N) = \sum _{\mathscr{N} (n) \leq N} d(n) $, and $1<p\leq
\infty$. To do so we study the averages
  $$ A_N f (x) = \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n) \,f(x-n)
,$$ and obtain improving and weighted maximal inequalities for our operator, in
the process.
