The group of isometries of the hyperbolic space ℍ³ is the 3-dimensional group PSL₂(ℂ), which is one of the simplest non-commutative complex Lie groups. Its quotient by the subgroup SO(3) ⊂ PSL₂(ℂ) naturally maps it back to ℍ³. Each fiber of this map is diffeomorphic to the real projective 3-space ℝℙ³. The resulting map PSL2(ℂ) → ℍ³ can be viewed as the simplest non-commutative counterpart of the map Log ∶ (ℂ^×)^𝑛 → ℝ^𝑛 from the commutative complex Lie group (ℂ^×)^𝑛 with the Lagrangian torus fibers that can be considered as a Liouville–Arnold type integrable system. Gelfand, Kapranov and Zelevinsky have introduced amoebas of algebraic varieties 𝑉 ⊂ (ℂ^×)^𝑛 as images Log(𝑉) ⊂ ℝ^𝑛. We define the amoeba of an algebraic subvariety of PSL₂(ℂ) as its image inℍ³. The paper surveys basic properties of the resulting hyperbolic amoebas and compares them against the commutative amoebas ℝ^𝑛.