Context. Even the spectroscopic capabilities of today's ground and space-based observatories can not keep up with the enormous flow of detections (>105 deg-2) unveiled in modern cosmological surveys as: i) would be required enormous telescope time to perform the spectroscopic follow-ups and ii) spectra remain unattainable for the fainter detected population. In the past decade, the typical accuracy of photometric redshift (photo-z) determination has drastically improved. Nowdays, it has become a perfect complement to spectroscopy, closing the gap between photometric surveys and their spectroscopic follow-ups. The photo-z precision for active galactic nuclei (AGN) has always lagged behind that for the galaxy population owing to the lack of proper templates and their intrinsic variability. Aims: Our goal is to characterize the ability of the Advanced Large, Homogeneous Area Medium-Band Redshift Astronomical (ALHAMBRA) survey in assigning accurate photo-z's to broad-line AGN (BLAGN) and quasi-stellar objects (QSOs) based on their ALHAMBRA very-low-resolution optical-near-infrared (NIR) spectroscopy. This will serve as a benchmark for any future compilation of ALHAMBRA selected QSOs and the basis for the statistical analysis required to derive luminosity functions up to z ~ 5. Methods: We selected a sample of spectroscopically identified BLAGN and QSOs and used a library of templates (including the SEDs of AGN and both normal and starburst galaxies, as well as stars) to fit the 23 photometric data points provided by ALHAMBRA in the optical and NIR (20 medium-band optical filters plus the standard JHKs). Results: We find that the ALHAMBRA photometry is able to provide an accurate photo-z and spectral classification for ~88% of the 170 spectroscopically identified BLAGN/QSOs over 2.5 deg2 in different areas of the survey and brighter than m678 = 23.5 (equivalent to rSLOAN ~ 24.0). The derived photo-z accuracy is below 1% and is comparable to the most recent results in other cosmological fields that use photometric information over a wider wavelength range. The fraction of outliers (~12%) is mainly caused by the larger photometric errors for the faintest sources and the intrinsic variability of the BLAGN/QSO population. A small fraction of outliers may have an incorrectly assigned spectroscopic redshift. Conclusions: The definition of the ALHAMBRA survey in terms of the number of filters, filter properties, areal coverage, and depth is able to provide photometric redshifts for BLAGN/QSOs with a precision similar to any previous survey that makes use of medium-band optical photometry. In agreement with previous literature results, our analysis also reveals that, in the 0 < z < 4 redshift interval, very accurate photo-z can be obtained without the use of NIR broadband photometry at the expense of a slight increase in the outliers. The importance of NIR data is expected to increase at higher z (z > 4). These results are relevant for the design of future optical follow-ups of surveys containing a large fraction of BLAGN, such as many X-ray or radio surveys. Based on observations collected at the German-Spanish Astronomical center, Calar Alto (Almeria, Spain), jointly operated by the Max-Planck-Institut für Astronomie at Heidelberg and the Instituto de Astrofísica de Andalucía (CSIC).