  We revisit the topic of power-free morphisms, focusing on the properties of
the class of complementary morphisms. Such morphisms are defined over a
$2$-letter alphabet, and map the letters 0 and 1 to complementary words. We
prove that every prefix of the famous Thue-Morse word $\mathbf{t}$ gives a
complementary morphism that is $3^+$-free and hence $\alpha$-free for every
real number $\alpha>3$. We also describe, using a 4-state binary finite
automaton, the lengths of all prefixes of $\mathbf{t}$ that give cubefree
complementary morphisms. Next, we show that $3$-free (cubefree) complementary
morphisms of length $k$ exist for all $k\not\in \{3,6\}$. Moreover, if $k$ is
not of the form $3\cdot2^n$, then the images of letters can be chosen to be
factors of $\mathbf{t}$. Finally, we observe that each cubefree complementary
morphism is also $\alpha$-free for some $\alpha<3$; in contrast, no binary
morphism that maps each letter to a word of length 3 (resp., a word of length
6) is $\alpha$-free for any $\alpha<3$.
  In addition to more traditional techniques of combinatorics on words, we also
rely on the Walnut theorem-prover. Its use and limitations are discussed.
