  The construction of the free Banach lattice generated by a real Banach space
is extended to the complex setting. It is shown that for every complex Banach
space $E$ there is a complex Banach lattice $FBL_{\mathbb C}[E]$ containing a
linear isometric copy of $E$ and satisfying the following universal property:
for every complex Banach lattice $X_{\mathbb C}$, every operator
$T:E\rightarrow X_{\mathbb C}$ admits a unique lattice homomorphic extension
$\hat{T}:FBL_{\mathbb C}[E]\rightarrow X_{\mathbb C}$ with $\|\hat{T}\|=\|T\|$.
The free complex Banach lattice $FBL_{\mathbb C}[E]$ is shown to have analogous
properties to those of its real counterpart. However, examples of
non-isomorphic complex Banach spaces $E$ and $F$ can be given so that
$FBL_{\mathbb C}[E]$ and $FBL_{\mathbb C}[F]$ are lattice isometric. The
spectral theory of induced lattice homomorphisms on $FBL_{\mathbb C}[E]$ is
also explored.
