  A complex unit gain graph ($ \mathbb{T} $-gain graph), $ \Phi=(G, \varphi) $
is a graph where the gain function $ \varphi $ assigns a unit complex number to
each orientation of an edge of $ G $ and its inverse is assigned to the
opposite orientation. The associated adjacency matrix $ A(\Phi) $ is defined
canonically. The energy $ \mathcal{E}(\Phi) $ of a $ \mathbb{T} $-gain graph $
\Phi $ is the sum of the absolute values of all eigenvalues of $ A(\Phi) $. For
any connected triangle-free $ \mathbb{T} $-gain graph $ \Phi $ with the minimum
vertex degree $ \delta$, we establish a lower bound $ \mathcal{E}(\Phi)\geq
2\delta$ and characterize the equality. Then, we present a relationship between
the characteristic and the matching polynomial of $ \Phi $. Using this, we
obtain an upper bound for the energy $ \mathcal{E}(\Phi)\leq
2\mu\sqrt{2\Delta_e+1} $ and characterize the classes of graphs for which the
bound sharp, where $ \mu$ and $ \Delta_e$ are the matching number and the
maximum edge degree of $ \Phi $, respectively. Further, for any unicyclic graph
$ G $, we study the gains for which the gain energy $ \mathcal{E}(\Phi) $
attains the maximum/minimum among all $ \mathbb{T} $-gain graphs defined on
$G$.
