The Eguchi-Hanson metric is a natural metric on the total space of the cotangent bundle \(T^*\mathbb{C}\mathbb{P}(1)\) of the complex projective line \(\mathbb{C}\mathbb{P}(1) \simeq \mathbb {S}^2\) , which extends the Fubini-Study metric of \(\mathbb{C}\mathbb{P}(1)\) . By virtue of the Mostow decomposition theorem, \(T^*\mathbb{C}\mathbb{P}(1)\) is isomorphic, as SU(2)-equivariant fiber bundle over \(\mathbb{C}\mathbb{P}(1)\) , to a complex (co-)adjoint orbit of \(SL(2, \mathbb {C})\) . In fact, this complex (co-)adjoint orbit is fibered over \(\mathbb{C}\mathbb{P}(1)\simeq \mathbb {S}^2\) with each fiber isomorphic to the hyperbolic disc \(\mathbb {H}^2\) . In this paper, we are interested in the complex structure inherited on the hyperbolic disc \(\mathbb {H}^2\) by the hyperkähler extension of the 2-sphere. Contrary to what is generally believed, we show that it differs from the natural complex structure of \(\mathbb {H}^2\subset \mathbb {C}\) inherited from its embedding in \(\mathbb {C}\) . In other words, the embedding of \(\mathbb {H}^2\) with its Hermitian-symmetric structure into the hyperkähler manifold \(T^*\mathbb{C}\mathbb{P}(1)\) is not holomorphic.