We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field.Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations.Next, we introduce two types of quaternionic algebras consisting of spaces of scalarvalued and quaternionic-valued functions. This is done using techniques that appear in the study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A ∞ -algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.We conclude the article with a discussion of relations between quaternionic analysis, representation theory of the conformal group, massless quantum electrodynamics and perspectives of further development of these subjects. and the fact that H is a division ring: Next we consider the algebra of complexified quaternions (also known as biquaternions) H C = C ⊗ R H and write elements of H C as Z = z 0 e 0 + z 1 e 1 + z 2 e 2 + z 3 e 3 , z 0 , z 1 , z 2 , z 3 ∈ C, so that Z ∈ H if and only if z 0 , z 1 , z 2 , z 3 ∈ R:Recall that we denote by S (respectively S ′ ) the irreducible 2-dimensional left (respectively right) H C -module, as described in Subsection 2.3 of [FL1]. The spaces S and S ′ can be realized as respectively columns and rows of complex numbers. ThenProof. The result follows from an observation that, for a fixed index l 0 , there are only finitely many non-zero coefficients a l 0 ,m,n l ′ ,m ′ ,n ′ , b l 0 ,m,n l ′ ,m ′ ,n ′ , c l 0 ,m,n l ′ ,m ′ ,n ′ , d l 0 ,m,n l ′ ,m ′ ,n ′ with that particular index, and similarly for index l ′ 0 .