"More important" is the comparative degree of important.
softer softest
more important, most important
happier, happiest
Technically, "completely" does not have any comparative form, because it is absolute and does not admit of degrees. Formally, "more completely" would be a suitable comparative, but for the reason already given does not make sense.
The comparative and superlative degrees of clean are cleaner and cleanest.
out
"More important" is the comparative degree of important.
Comparative = more important Superlative = most important
More important is the comparative form, and most important is the superlative form.
more in, most in
more out, most out
more in, most in
The three degrees (of comparison) for adjectives are Positive, Comparative and Superlative. Example: hard (positive) harder (comparative) hardest (superlative)
Comparative and superlative degrees are for adjectives and adverbs. House can be used as a noun or a verb and does not have comparative or superlative forms.
greater & greatest
merrier, merriest