There are no "strange implications"
Regardless of which interpretation of "at any time" you use (whether it can it be done when it is not your own turn) no problems arise from the alter self spell. This is because the section on combat states:
On your turn, you can move a distance up to your speed and take one action.
This explicitly limits the times that you can take an action to on your own turn. And so if it is not your turn you cannot take the action. "At any time" does not change situations where you couldn't take the action anyhow.
Similarly then, "at any time" does not somehow let you take this action even while incapacitated or otherwise prevented from taking actions.
The phrase, honestly, seems to be completely filler from a strict RAW reading which makes sense because dnd-5e is written in standard English so there are bound to technically unnecessary phrases and words throughout its rules.You can't use alter self on somebody else's turn because you can't take actions then.
As user @PJRZ pointed out in a comment, you can simply remove the phrase and nothing is lost:
For the duration of the spell, you can use your action to change your appearance in this way again.
But when is "at any time"?
See the question "When exactly can a caster stop concentrating on a spell?" which already has different answers providing different points of view about whether "at any time" includes during somebody else's turn. The (by far) most upvoted answer there states that "any time" does include other people's turns.
If something can be done "At any time" then it can be done "at any time" that you otherwise could do that thing. It doesn't magically remove restrictions on the timing, otherwise you could do things like use alter self even while incapacitated, indeed a strange implication. If something can be done "at any time" you still must follow any timing restrictions that come from other rules (such as in the case of alter self)
The debate here is whether there exists some rule that prevents you from doing things on somebody else's turn and different answers to that question can be seen on the answers to the question "When exactly can a caster stop concentrating on a spell?".
This means you technically have four options:
- "At any time" overrides the rule that prevents you from taking non-actions on another's turn but does not override the one that prevents you from taking actions on another's turn.
- "At any time" overrides all rules so you can use alter self even while incapacitated.
- "At any time" does not override any rules and there is not a rule which prevents you from taking non-actions on somebody else's turn.
- "At any time" does not override any rules and there is a rule which prevents you from taking non-actions on somebody else's turn.
For discussion on whether there is a rule preventing you from taking non-actions on another's turn see the question "When can you do things that do not require an action?" and various related questions linked there.