Minimal Surfaces via Complex Quaternions

I will present a quaternionic description of minimal surfaces in conformal parametrization. In the classical setting, minimal surfaces arise from holomorphic null curves through the Weierstrass--Enneper representation. The key observation is that the nullity condition can be encoded very compactly using complexified quaternions, in the form
\[
\Phi(z)=\chi(z)\,L\,\chi(z)^{-1}, \qquad L=i+\sqrt{-1}\,j.
\]
This viewpoint makes the underlying algebraic structure explicit and reveals a natural analogy with Pythagorean-hodograph curves, where quaternionic factorizations also play a central role. I will discuss some standard examples and briefly explain the appearance of Sylvester-type equations in explicit computations.