As a base for creating nonlinear algebra, we use vectors — mathematical objects with some predefined general properties, but without defining these objects in terms of numbers or numerical matrices. Next, we build our algebra based on the vector multiplication operation. Our approach allows us to obtain new and more generalized conceptions of vectors, scalars and related objects of a mixed scalar-vector type — generalized quaternions. We propose a fundamentally new perspective on such familiar concepts as space, vectors, quaternions, complexity, parallelism, orthogonality, and dimension. Within the framework of new nonlinear algebra, geometric concepts such as parallelism and orthogonality acquire the operator meaning of vector commutativity and anticommutativity. The essence of the transition from linear to nonlinear representations lies in the transition from static geometric representations to operational ones. Vector cycles, which are triples of vectors cyclically connected to each other, occupy a special place in our algebraic system. The axiomatic framework we have constructed allows us to prove a number of statements important for the further development of the theory of nonlinear spaces.