  The Bianchi groups ${\rm Bi}(d)={\rm PSL}(2,\mathcal{O}_d) < {\rm PSL}(2,\C)$
(where $\mathcal{O}_d$ denotes the ring of integers of $\Q (i\sqrt{d})$, with
$d \geqslant 1$ squarefree) can be viewed as subgroups of ${\rm SO}(3,1)$ under
the isomorphism ${\rm PSL}(2,\C) \simeq {\rm SO}^0(3,1)$. We study the
deformations of these groups into the larger Lie groups ${\rm SU}(3,1)$ and
${\rm SL}(4,\R)$ for small values of $d$. In particular we show that ${\rm
Bi}(3)$, which is rigid in ${\rm SO}(3,1)$, admits a 1-dimensional deformation
space into ${\rm SU}(3,1)$ and ${\rm SL}(4,\R)$, whereas any deformation of
${\rm Bi}(1)$ into ${\rm SU}(3,1)$ or ${\rm SL}(4,\R)$ is conjugate to one
inside ${\rm SO}(3,1)$. We also show that none of the deformations into ${\rm
SU}(3,1)$ are both discrete and faithful.
