  The powerful concept of an operator ideal on the class of all Banach spaces
makes sense in the real and in the complex case. In both settings we may, for
example, consider compact, nuclear, or $2$--summing operators, where the
definitions are adapted to each other in a natural way. This paper deals with
the question whether or not that fact is based on a general philosophy. Does
there exists a one--to--one correspondence between ``real properties'' and
``complex properties'' defining an operator ideal? In other words, does there
exist for every real operator ideal a uniquely determined corresponding complex
ideal and vice versa?
  Unfortunately, we are not abel to give a final answer. Nevertheless, some
preliminary results are obtained. In particular, we construct for every real
operator ideal a corresponding complex operator ideal and for every complex
operator ideal a corresponding real one. However, we conjecture that there
exists a complex operator ideal which can not be obtained from a real one by
this construction.
  The following approach is based on the observation that every complex Banach
space can be viewed as a real Banach space with an isometry acting on it like
the scalar multiplication by the imaginary unit $i$.
