  This paper extends the concept of a $B$-happy number, for $B \geq 2$, from
the rational integers, $\mathbb{Z}$, to the Gaussian integers, $\mathbb{Z}[i]$.
We investigate the fixed points and cycles of the Gaussian $B$-happy functions,
determining them for small values of $B$ and providing a method for computing
them for any $B \geq 2$. We discuss heights of Gaussian $B$-happy numbers,
proving results concerning the smallest Gaussian $B$-happy numbers of certain
heights. Finally, we prove conditions for the existence and non-existence of
arbitrarily long arithmetic sequences of Gaussian $B$-happy numbers.
