Applied Physics and Mathematics Annotation << Back WHAT EXACTLY IS A PHOTON? THE NATURE OF RADIATION QUANTA AND COSMOLOGICAL REDSHIFT. PART 2 A.V. KUKUSHKIN Part 2 of the article is also preparatory like Part 1. Here, I continue to analyze the consequences arising from the introduction of a new value to mathematics and therefore to the mathematical formalism of special relativity, the imaginary directional unit (an imaginary vector). As a result of this introduction, the 4-vectors in Minkowski space become complex (C-vectors). Their algebraic nature differs from that of C-vectors in the complex plane. In Part 1 of the article, a definition and the rules for adding their real part to the imaginary one was proposed for such C-vectors. As a consequence of these rules, any C-vector in the Minkowski space is equivalent (or twinned) to either an ordinary real or an imaginary or zero (isotropic) vector. It follows that when one inertial frame of reference moves relative to another, the coordinate axes of the first one do not experience rotation. Due to this and to the relativistic time dilation the laws of Lorentz kinematics in the complex 4D space turn out to be fully compatible with the laws of Galilean kinematics in the real 3D space. In the article, the Lorentz transformations for the basis vectors of a generalized curvilinear and orthogonal frame of reference of cylindrical type and transformations for non-unit vectors of the covariant basis of these systems are derived. Those will be needed for the further modeling of the full photon wave function. Keywords: magnitude of the imaginary unit vector, complex Minkowski space, imaginary vector, complex vector. DOI: 10.25791/pfim.05.2025.1341 Pp. 06-27.