Almost complex connections on almost complex manifolds with Norden metric

A four-parametric family of linear connections preserving the almost complex structure is defined on an almost complex manifold with Norden metric. Necessary and sufficient conditions for these connections to be natural are obtained. A two-parametric family of complex connections is studied on a conformal K\"{a}hler manifold with Norden metric. The curvature tensors of these connections are proved to coincide.