  In this paper, we investigate the existence of Ulrich bundles on a smooth
complete intersection of two $4$-dimensional quadrics in $\mathbb P^5$ by two
completely different methods. First, we find good ACM curves and use Serre
correspondence in order to construct Ulrich bundles, which is analogous to the
construction on a cubic threefold by Casanellas-Hartshorne-Geiss-Schreyer.
Next, we use Bondal-Orlov's semiorthogonal decomposition of the derived
category of coherent sheaves to analyze Ulrich bundles. Using these methods, we
prove that any smooth intersection of two 4-dimensional quadrics in $\mathbb
P^5$ carries an Ulrich bundle of rank $r$ for every $r \ge 2$. Moreover, we
provide a description of the moduli space of stable Ulrich bundles.
