Quantifying integrated information is a leading approach towards building a fundamental theory of consciousness. Integrated Information Theory (IIT) has gained attention in this regard due to its theoretically strong framework. However, it faces some limitations such as current state dependence, computationally expensive and inability to be applied to real brain data. On the other hand, Perturbational Complexity Index (PCI) is a clinical measure for distinguishing different levels of consciousness. Though PCI claims to capture the functional differentiation and integration in brain networks (similar to IIT), its link to integrated information theories is rather weak. Inspired by these two approaches, we propose a new measure - $Φ^C$ using a novel compression-complexity perspective that serves as a bridge between the two, for the first time. $Φ^C$ is founded on the principles of lossless data compression based complexity measures which characterize the dynamical complexity of brain networks. $Φ^{C}$ exhibits following salient innovations: (i) mathematically well bounded, (ii) negligible current state dependence unlike $Φ$, (iii) integrated information measured as compression-complexity rather than as an infotheoretic quantity, and (iv) faster to compute since number of atomic bipartitions scales linearly with the number of nodes of the network, thus avoiding combinatorial explosion. Our computer simulations show that $Φ^C$ has similar hierarchy to $<Φ>$ for several multiple-node networks and it demonstrates a rich interplay between differentiation, integration and entropy of the nodes of a network. $Φ^C$ is a promising heuristic measure to characterize the quantity of integrated information (and hence a measure of quantity of consciousness) in larger networks like human brain and provides an opportunity to test the predictions of brain complexity on real neural data.