An example in $GL(3, {\mathbb C})$ was first given by Bass (answering a question by Kaplansky) in Example 1.10 of

Bass, Hyman, Groups of integral representation type, Pac. J. Math. 86, 15-51 (1980). ZBL0444.20006.

In section 1 of his paper Bass also discusses the general structure of subgroups of $GL(n, {\mathbb C})$ where every element is unitarizable.

Here is the example. Start with a free subgroup $F=\langle s, t\rangle$ of rank 2 in $SU(2)$ acting linearly on ${\mathbb C}^2$ with generators acting as matrices $A, B$. Now, deform $F$ to a free group of affine transformations by $$ \rho(s)=A, \rho(t)x= Bx+ v, $$ where $v$ is any nonzero vector in ${\mathbb C}^2$. Then each element of $G=\rho(F_2)$ is affine-conjugate to its linear part, an element of $SU(2)$. At the same time, $G$ has no fixed points in the complex-affine space ${\mathbb C}^2$. Lastly, use the fact that the group of complex-affine transformations of ${\mathbb C}^2$ embeds in $GL(3, {\mathbb C})$ sending an affine transformation $$ x\mapsto Mx + v $$ to the matrix $$ \left[\begin{array}{cc} M&v\\ 0&1 \end{array}\right]. $$ This gives an example of a non-unitarizable subgroup of $SL(3, {\mathbb C})$ such that each individual element is unitarizable.

An example in $GL(3, {\mathbb C})$ was first given by Bass (answering a question by Kaplansky) in Example 1.10 of

Bass, Hyman, Groups of integral representation type, Pac. J. Math. 86, 15-51 (1980). ZBL0444.20006.

In section 1 of his paper Bass also discusses the general structure of subgroups of $GL(n, {\mathbb C})$ where every element is unitarizable.

Here is the example. Start with a free subgroup $F=\langle s, t\rangle$ of rank 2 in $SU(2)$ acting linearly on ${\mathbb C}^2$ with generators acting as matrices $A, B$. Now, deform $F$ to a free group of affine transformations by $$ \rho(s)=A, \rho(t)x= Bx+ v, $$ where $v$ is any nonzero vector in ${\mathbb C}^2$. Then each element of $G=\rho(F_2)$ is affine-conjugate to its linear part, an element of $SU(2)$. At the same time, $G$ has no fixed points in the complex-affine space ${\mathbb C}^2$. It follows that $G$ is unbounded. Lastly, use the fact that the group of complex-affine transformations of ${\mathbb C}^2$ embeds in $GL(3, {\mathbb C})$ sending an affine transformation $$ x\mapsto Mx + v $$ to the matrix $$ \left[\begin{array}{cc} M&v\\ 0&1 \end{array}\right]. $$ Under this map unbounded subsets map to unbounded subsets. This gives an example of a non-unitarizable subgroup of $SL(3, {\mathbb C})$ such that each individual element is unitarizable.

An example in $GL(4, {\mathbb C})$ was first given in

Bass, Hyman, Groups of integral representation type, Pac. J. Math. 86, 15-51 (1980). ZBL0444.20006.

However, the complex dimension can be lowered to 3 (which is the lowest possible). Unless I am mistaken, this example appears in the paper by Kulkarni (to which I do not have access at the moment):

Kulkarni, Ravi S., Conjugacy classes in M(n), Conformal geometry, Semin., MPI, Bonn/FRG 1985/86, Aspects Math.: E, 12, 41-64 (1988). ZBL0659.53014.

Here is the example. Start with a free subgroup $F=\langle s, t\rangle$ of rank 2 in $SU(2)$ acting linearly on ${\mathbb C}^2$ with generators acting as matrices $A, B$. Now, deform $F$ to a free group of affine transformations by $$ \rho(s)=A, \rho(t)x= Bx+ v, $$ where $v$ is any nonzero vector in ${\mathbb C}^2$. Then each element of $G=\rho(F_2)$ is affine-conjugate to its linear part, an element of $SU(2)$. At the same time, $G$ has no fixed points in the complex-affine space ${\mathbb C}^2$. Lastly, use the fact that the group of complex-affine transformations of ${\mathbb C}^2$ embeds in $GL(3, {\mathbb C})$ sending an affine transformation $$ x\mapsto Mx + v $$ to the matrix $$ \left[\begin{array}{cc} M&v\\ 0&1 \end{array}\right]. $$ This gives an example of a non-unitarizable subgroup of $SL(3, {\mathbb C})$ such that each individual element is unitarizable.

An example in $GL(3, {\mathbb C})$ was first given by Bass (answering a question by Kaplansky) in Example 1.10 of

Bass, Hyman, Groups of integral representation type, Pac. J. Math. 86, 15-51 (1980). ZBL0444.20006.

In section 1 of his paper Bass also discusses the general structure of subgroups of $GL(n, {\mathbb C})$ where every element is unitarizable.

Here is the example. Start with a free subgroup $F=\langle s, t\rangle$ of rank 2 in $SU(2)$ acting linearly on ${\mathbb C}^2$ with generators acting as matrices $A, B$. Now, deform $F$ to a free group of affine transformations by $$ \rho(s)=A, \rho(t)x= Bx+ v, $$ where $v$ is any nonzero vector in ${\mathbb C}^2$. Then each element of $G=\rho(F_2)$ is affine-conjugate to its linear part, an element of $SU(2)$. At the same time, $G$ has no fixed points in the complex-affine space ${\mathbb C}^2$. Lastly, use the fact that the group of complex-affine transformations of ${\mathbb C}^2$ embeds in $GL(3, {\mathbb C})$ sending an affine transformation $$ x\mapsto Mx + v $$ to the matrix $$ \left[\begin{array}{cc} M&v\\ 0&1 \end{array}\right]. $$ This gives an example of a non-unitarizable subgroup of $SL(3, {\mathbb C})$ such that each individual element is unitarizable.