Atomic systems, such as trapped atomic ions, are a leading candidate for quantum computing. Over the past few decades, researchers have developed unprecedented coherent control over quantum two-level systems (qubits) encoded in these systems. The focus on two-level systems has neglected many other internal quantum states in these atoms, which we are now exploring to increase performance and enable new capabilities. In this talk, I will provide an overview of the trapped-ion research performed at MIT and MIT Lincoln Laboratory, focusing in particular on research utilizing the extended Hilbert spaces available in these systems. I will discuss our work using the internal states of calcium ions for quantum signal processing algorithms, quantum sensing, and for encoding an error-corrected logical qubit entirely within a single ion.