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