Independent trees are used in building secure and/or fault-tolerant network communication protocols. They have been investigated for different network topologies including tori. Dense Gaussian networks are potential alternatives for 2-dimensional tori. They have similar topological properties; however, they are superiors in carrying communications due to their node-distance distributions and smaller diameters. In this paper, we present constructions of edge-disjoint node-independent spanning trees in dense Gaussian networks. Based on the constructed trees, we design algorithms that could be used in fault-tolerant routing or secure message distribution.