Summary: 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 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 t that give cubefree complementary morphisms. Next, we show that 3-free (cubefree) complementary morphisms of length \(k\) exist for all \(k \notin \{3, 6\}\). Moreover, if \(k\) is not of the form \(3 \cdot 2^n\), then the images of letters can be chosen to be factors of 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.