Flip-flop processes refer to a family of stochastic fluid processes which converge to either a standard Brownian motion (SBM) or to a Markov modulated Brownian motion (MMBM). In recent years, it has been shown that complex distributional aspects of the univariate SBM and MMBM can be studied through the limiting behaviour of flip-flop processes. Here, we construct two classes of bivariate flip-flop processes whose marginals converge strongly to SBMs and are dependent on each other, which we refer to as \emph{alternating} two-dimensional Brownian motion processes}. While the limiting bivariate processes are not Gaussian, they possess desirable qualities, such as being tractable and having a time-varying correlation coefficient function.