  This paper is devoted to study multiplicity and regularity as well as to
present some classifications of complex analytic sets. We present an
equivalence for complex analytical sets, namely blow-spherical equivalence and
we receive several applications with this new approach. For example, we reduce
to homogeneous complex algebraic sets a version of Zariski's multiplicity
conjecture in the case of blow-spherical homeomorphism, we give some partial
answers to the Zariski's multiplicity conjecture, we show that a blow-spherical
regular complex analytic set is smooth and we give a complete classification of
complex analytic curves.
