  Conjucyclic codes are an important and special family of classical
error-correcting codes, which have been used to construct binary quantum
error-correcting codes (QECCs). However, at present, the research on the
conjucyclic codes is extremely insufficient. This paper will explore the
algebraic structure of additive conjucyclic codes over $\mathbb{F}_{q^{2}}$ for
the first time. Mainly via the trace function from $\mathbb{F}_{q^{2}}$ down
$\mathbb{F}_{q}$, we will firstly build an isomorphic mapping between $q^2$-ary
additive conjucyclic codes and $q$-ary linear cyclic codes. Since the mapping
preserves the weight and orthogonality, then the dual structure of these codes
with respect to the alternating inner product will be described. Then a new
construction of QECCs from conjucyclic codes can be obtained. Finally, the
enumeration of $q^2$-ary additive conjucyclic codes of length $n$ and the
explicit forms of their generator and parity-check matrices will be determined.
