We adopt a projective viewpoint to show that any holomorphic function \(h:\mathbb {C}^{n}\rightarrow \mathbb {C}\) can be regarded as the “support function” of a “complex hedgehog” \(\mathcal {H}_{h}\) , defined by a holomorphic parametrization \(x_{h}:\mathbb {C}^{n}\rightarrow \mathbb {C}^{n+1}\) . We introduce the notion of the evolute of \(\mathcal {H}_{h}\) in \(\mathbb {C}^{2}\) , and a new natural notion of complex curvature, interpreting the evolute as the locus of the centers of complex curvature. On the unit disc \(\mathbb {D}\) of \(\mathbb {C}\) , the space of holomorphic functions up to a similitude admits a scalar product which can be interpreted as a mixed symplectic area. We give a sharp estimate of the (symplectic) area of \(x_{h}(\mathbb {D})\) . Returning to real hedgehogs in \(\mathbb {R}^{2n}\) with a linear complex structure, we study their evolutes, focusing on \(\mathbb {R}^{4}\) with a linear Kähler structure determined by a pure unit quaternion. We also study the symplectic area of the images of oriented Hopf circles under hedgehog parametrizations and introduce a quaternionic curvature function. Finally, we consider hedgehogs in \(\mathbb {R}^{4n}\) regarded as a hyperkähler vector space.