Context. It is debated whether the Milky Way bulge has characteristics more similar to those of a classical bulge than those of a pseudobulge. Detailed abundance studies of bulge stars are important when investigating the origin, history, and classification of the bulge. These studies provide constraints on the star-formation history, initial mass function, and differences between stellar populations. Not many similar studies have been completed because of the large distance and high variable visual extinction along the line-of-sight towards the bulge. Therefore, near-IR investigations can provide superior results. Aims. To investigate the origin of the bulge and study its chemical abundances determined from near-IR spectra for bulge giants that have already been investigated with optical spectra. The optical spectra also provide the stellar parameters that are very important to the present study. In particular, the important CNO elements are determined more accurately in the near-IR. Oxygen and other α elements are important for investigating the star-formation history. The C and N abundances are important for determining the evolutionary stage of the giants and the origin of C in the bulge. Methods. High-resolution, near-infrared spectra in the H band were recorded using the CRIRES spectrometer mounted on the Very Large Telescope. The CNO abundances are determined from the numerous molecular lines in the wavelength range observed. Abundances of the α elements Si, S, and Ti are also determined from the near-IR spectra. Results. The abundance ratios [O/Fe], [Si/Fe], and [S/Fe] are enhanced to metallicities of at least [Fe/H] = −0.3, after which they decline. This suggests that the Milky Way bulge experienced a rapid and early burst of star formation similar to that of a classical bulge. However, a similarity between the bulge trend and the trend of the local thick disk seems to be present. This similarity suggests that the bulge could have had a pseudobulge origin. The C and N abundances suggest that our giants are first-ascent red-giants or clump stars, and that the measured oxygen abundances are those with which the stars were born. Our [C/Fe] trend does not show any increase with [Fe/H], which is expected if W-R stars contributed substantially to the C abundances. No "cosmic scatter" can be traced around our observed abundance trends: the measured scatter is expected, given the observational uncertainties.