The behavior of complex truncated Navier-Stokes equations is investigated. A significant role in the phenomenology of these equations is played by a set of infinitely many invariant N-dimensional hyperplanes (subspaces of the 2N-dimensional phase space) which are symmetrically placed by virtue of a symmetry group. There is numerical evidence that, at least for Reynolds numbers below a certain critical threshold, every random initial point is captured first by one of these hyperplanes, then by an attractor on it. Particular bifurcations can take place in the 2N-phase space. A fixed point can bifurcate into a closed curve consisting entirely of fixed points; in addition, a closed curve of fixed points can bifurcate into a torus covered entirely by either fixed points or periodic orbits. As long as the invariant hyperplanes are attracting, a complete correspondence is found between the phenomenologies of some complex models and their previously studied real versions.