Streaming Big Data applications require the means to efficiently utilize large-scale data in an online manner. This issue becomes even more pressing when data are also multidimensional, as is the case with quaternion data streams. To this end, we first introduce the online censoring (OC) based quaternion least mean square (OC-QLMS) and OC-augmented QLMS (OC-AQLMS) algorithms, which censor less informative data in order to reduce computational complexity without severely affecting performance. Next, to censor both the outlier and noninformative data, we also propose the robust OC-QLMS (ROC-QLMS) and ROC-AQLMS. Fixed and adaptive threshold rules are introduced into the proposed OC algorithms to efficiently implement the desired censoring probability in the quaternion domain. The fundamental convergence analysis on the step size for all the proposed algorithms is also presented and the superior properties of the proposed algorithms are demonstrated in system identification scenarios.