Consider a one-dimensional, non-relativistic particle with a two state internal degree of freedom called "spin".
Restricting to the bosonic or fermionic sectors gives two partner potentials determined by In supersymmetric quantum field theories with four spacetime dimensions, which might have some connection to nature, it turns out that scalar fields arise as the lowest component of a chiral superfield, which tends to automatically be complex valued.
In this context, holomorphic means that the function can only depend on the chiral superfields, not their complex conjugates.
The fact that W is holomorphic in the chiral superfields helps explain why supersymmetric theories are relatively tractable, as it allows one to use powerful mathematical tools from complex analysis.
Note that non-perturbative processes may correct this, for example through contributions to the beta functions due to instantons.