  Complex periodic variables (CPVs) are stars that exhibit highly structured
and periodic optical light curves. Previous studies have indicated that these
stars are typically disk-free pre-main-sequence M dwarfs with rotation periods
ranging from 0.2 to 2 days. To advance our understanding of these enigmatic
objects, we conducted a blind search using TESS 2-minute data of 65,760 K and M
dwarfs with $T$<16 mag and $d$<150 pc. We found 50 high-quality CPVs, and
subsequently determined that most are members of stellar associations. Among
the new discoveries are the brightest ($T$$\approx$9.5 mag), closest
($d$$\approx$20 pc), and oldest ($\approx$200 Myr) CPVs known. One exceptional
object, LP 12-502, exhibited up to eight flux dips per cycle. Some of these
dips coexisted with slightly different periods, and the shortest-duration dips
precisely matched the expected timescale for transiting small bodies at the
corotation radius. Broadly, our search confirms that CPVs are mostly young
($\lesssim$150 Myr) and low-mass ($\lesssim$0.4 $M_\odot$). The flux dips
characteristic of the class have lifetimes of $\approx$100 cycles, although
stellar flares seem to induce sudden dip collapse once every few months. The
most plausible explanation for these phenomena remains corotating
concentrations of gas or dust. The gas or dust is probably entrained by the
star's magnetic field, and the sharp features could result from a multipolar
field topology, a hypothesis supported by correspondences between the light
curves of CPVs and of rapidly rotating B stars known to have multipolar
magnetic fields.
