High extinction and crowding create a natural limitation for optical surveys towards the central regions of the Milky Way, where the gas and dust are mainly confined. Large-scale near-infrared (IR) surveys of the Galactic plane and bulge are a good opportunity to explore open scientific questions as well as to test our capability to explore future data sets efficiently. Thanks to the VISTA Variables in the Vía Láctea (VVV) ESO public survey, it is now possible to explore a large number of objects in those regions. This paper addresses the variability analysis of all VVV point sources having more than 10 observations in VVVDR4 using a novel approach. In total, the near-IR light curves of 288 378 769 sources were analysed using methods developed in the New Insight Into Time Series Analysis project. As a result, we present a complete sample having 44 998 752 variable star candidates (VVV-CVSC), which include accurate individual coordinates, near-IR magnitudes (Z,Y,J, and Hs), extinctions A(Ks), variability indices, periods, amplitudes, among other parameters to assess the science. Unfortunately, a side effect of having a highly complete sample, is also having a high level of contamination by non-variable (contamination ratio of non-variables to variables is slightly over 10:1). To deal with this, we also provide some flags and parameters that can be used by the community to decrease the number of variable candidates without heavily decreasing the completeness of the sample. In particular, we cross-identified 339 601 of our sources with Simbad and AAVSO data bases, which provide us with information for these objects at other wavelengths. This subsample constitutes a unique resource to study the corresponding near-IR variability of known sources as well as to assess the IR variability related with X-ray and gamma-ray sources. On the other hand, the other ∼99.5 per cent sources in our sample constitutes a number of potentially new objects with variability information for the heavily crowded and reddened regions of the Galactic plane and bulge. The present results also provide an important queryable resource to perform variability analysis and to characterize ongoing and future surveys like TESS and LSST.