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When I first looked at what I call Amphibian Chess I had just viewed George
P. Jelliss's article on chess variants and found his description of 'Sea
Chess' (line pieces capture by vaulting to next space beyond). The term
'Sea pieces' was new to me (I was familiar with the 'Mermaid' or Q
version only), and I was struck with the practical flaw in the game:
pieces on the edge were relatively difficult to capture. Since the 'sea
pieces' were in contrast to the more usual FIDE (land?) pieces and
'amphibian' pieces seemed like an amusing compromise, I toyed with how
these pieces might work.
 
I looked at this as a 'thematic' task, and I usually focus on making
only the necessary changes to produce an interesting and (possibly)
playable game (Occam's razor, or something like that). The sea-land
compromise was obvious for line pieces, but Knights presented a problem.
Three possible solutions were considered:

(a) leave the standard Knight alone
(b) define a 'space beyond' to correspond to a second Knight's leap
(c) introduce a devise to break down the Knight's move

Choice (a) seemed unworthy of something called 'Amphibian Chess'. Choice
(b) was probably the most elegant option and it's playable, but it seems
to makes the 8x8 board seem pretty small. Instead, I chose choice (c) and
adopted the Chinese 'Mao' move.

Since Peter seemed to be addressing a similar problem, I added my comment.
The approach admits to several generalizations (e.g., other Amphibian
pieces) that you may wish to explore. I hope you and other CV enthusiasts
will do so.

BTW my 'rules' left out the qualification that the 'next space beyond'
should be vacant. (Sorry. Rule writing is a tough job.)