  The applications of additive codes mainly lie in quantum error correction and
quantum computing. Due to their applications in quantum codes, additive codes
have grown in importance. In addition to this, additive codes allow the
implementation of a variety of dualities. The article begins by developing the
properties of Additive Complementary Dual (ACD) codes with respect to arbitrary
dualities over finite abelian groups. Further, we introduce a subclass of
non-symmetric dualities referred to as skew-symmetric dualities. Then, we
precisely count symmetric and skew-symmetric dualities over finite fields. Two
conditions have been obtained: one is a necessary and sufficient condition, and
the other is a necessary condition. The necessary and sufficient condition is
for an additive code to be an ACD code over arbitrary dualities. The necessary
condition is on the generator matrix of an ACD code over skew-symmetric
dualities. We provide bounds for the highest possible minimum distance of ACD
codes over skew-symmetric dualities. Finally, we find some new quaternary ACD
codes over non-symmetric dualities with better parameters than the symmetric
ones.
