We study the complex spectrum of the partial theta function $$\Theta(q,x)=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j, \qquad |q|<1,$$ where a spectral value is a parameter for which $\Theta(q,\cdot)$ has a multiple zero. Since the function is defined here only for $|q|<1$, all spectral values are strictly inside the unit disk; boundary points on $|q|=1$ occur only as accumulation points of the spectrum. The paper combines two complementary points of view. Near the unit circle we prove that every point of $|q|=1$ is an accumulation point of the spectrum; the proof uses explicit spectral factors of truncations, the Jacobi triple product, and a boundary-window lifting argument near roots of unity. Inside a fixed subdisk, illustrated for $|q|\leq 0.8$, the true spectrum is locally finite and must be separated carefully from the much larger branch loci of truncations and Jensen polynomials. We give a truncation-seeded Newton procedure which produces a discrete list of candidate spectral values, explain the caustic/escaping-root mechanism in finite approximants, and record numerical monodromy experiments for the first dominating first-quadrant spectral points, for the second layer, and for the first negative real spectral points, with a separate warning about the natural base point on the negative real axis.