Not since 1987 has the electron magnetic moment and the fine structure constant been measured more accurately. Now, a one-electron quantum cyclotron has made possible much more accurate measurements of both. The dimensionless electron magnetic moment (often called the electron g value) is measured 15 times more accurately than in a celebrated measurement that stood for 20 years. The fine structure constant is measured 20 times more accurately than in any independent measurement. A quantum non-demolition measurement reveals the quantum structure in the cyclotron motion of an electron suspended by itself for months at a time. Cavity-inhibited spontaneous emission and a one-particle self-excited oscillator (SEO) give the resolution needed to carry out quantum-jump spectroscopy of the lowest energy levels of the weakly-bound electron-apparatus system. The SEO is the classical measurement system for the quantum states of the electron cyclotron motion and spin.