  A Diophantine $m$-tuple is a set of $m$ distinct integers such that the
product of any two distinct elements plus one is a perfect square. In this
paper we study the extensibility of a Diophantine triple $\{k-1, k+1,
16k^3-4k\}$ in Gaussian integers $\mathbb{Z}[i]$ to a Diophantine quadruple.
Similar one-parameter family, $\{k-1, k+1, 4k\}$, was studied in
Franu\v{s}i\'c's previous paper, where it was shown that the extension to a
Diophantine quadruple is unique (with an element $16k^3-4k$). The family of the
triples of the same form $\{k-1, k+1, 16k^3-4k\}$ was already studied in
rational integers. It appeared as a special case while solving the
extensibility problem of Diophantine pair $\{k-1, k+1\}$, in which it was not
possible to use the same method as in the other cases. As authors (Bugeaud,
Dujella and Mignotte) point out, the difficulty appears because the gap between
$k+1$ and $16k^3-4k$ is not sufficiently large. We find the same difficulty
here while trying to use Diophantine approximations. Then we partially solve
this problem by using linear forms in logarithms.
