  We propose a conjecture extending the classical construction of elliptic
units to complex cubic number fields $K$. The conjecture concerns special
values of the elliptic gamma function, a holomorphic function of three complex
variables arising in mathematical physics whose transformation properties under
$\mathrm{SL}_3(\mathbf{Z})$ were studied by Felder and Varchenko in the early
2000s. Using this function we construct complex numbers that we conjecture to
be units in narrow ray class fields of $K$. We also propose a reciprocity law
for the action of the Galois group on these units in the style of Shimura.
  To support our conjecture we offer numerical evidence and also prove a new
type of Kronecker limit formula relating the logarithm of the modulus of these
complex numbers to the derivatives at $s = 0$ of partial zeta functions of $K$.
Our constructions unveil the role played by the elliptic gamma function in
Hilbert's twelfth problem for complex cubic fields.
