Advances in experimental techniques and data collection have fueled the rise of the complex systems perspective throughout biology. The complex systems approach appreciates the entirety of the system; focusing not on individual units but on how the collection of pieces together, in context, give rise to function. This perspective informs analysis directions, but in and of itself does not naturally invoke a specific methodological mate. In this work we argue that algebraic topology is an ideological match as a framework for the complex systems perspective, because topology by nature represents, operates on, and involves the system totality. We demonstrate the ability of topology to provide insight into biological and linguistic system structures, and offer suggestions for further method development. Finally we close with a discussion of the frontiers of topological data analysis, which we believe will indeed shape the future of research in complex biological systems.