Magnetic particle imaging (MPI) is an imaging modality exploiting the nonlinear magnetization behavior of (super-)paramagnetic nanoparticles to obtain a space- and often also time-dependent concentration of a tracer consisting of these nanoparticles. MPI has a continuously increasing number of potential medical applications. One prerequisite for successful performance in these applications is a proper solution to the image reconstruction problem. More classical methods from inverse problems theory, as well as novel approaches from the field of machine learning, have the potential to deliver high-quality reconstructions in MPI. We investigate a novel reconstruction approach based on a deep image prior, which builds on representing the solution by a deep neural network. Novel approaches, as well as variational and iterative regularization techniques, are compared quantitatively in terms of peak signal-to-noise ratios and structural similarity indices on the publicly available Open MPI dataset.