When you end your movement, you can collect all the clue tokens where you end up.

If an encounter moves you to another location, does that count as ending your movement? Or does ending your movement actually mean "at the end of the movement phase"?

Nope, if an encounter sends you to another location, you don't deal with the monsters there (or pick up any of the clues? I forgot that this was not official, but it is certainly implied). If there's a gate there, then your encounter is to enter it, even if the encounter that sent you there said to draw X encounters and pick one .

snacknuts said:

From pages 7 and 8.

"Picking Up Clues
Any time an investigator ends his movement in a location
that contains Clue tokens, he may immediately take
any or all of those Clue tokens. The investigator may not
take any Clue tokens if he merely moves through the
location and then continues his movement: He must end
his movement in the location containing the Clue tokens."

Ending Movement does it.

Once your Arkham Encounter phase begins, your movement and movement phase is defintely over.

What's not clear to me is whether you have to engage monsters before you pick up the clues. Do you end movement and enter into combat, for example, or does the actual combat end Movement. In other words is "ending Movement" and "ending the Movement Phase" the same thing?

The prior paragraph reads this way:

"Once an investigator begins combat with a monster for
any reason, his movement is over. Regardless of whether
or not he wins the battle, the investigator loses the rest
of his movement points and must remain where he is."

It seems to me that "his movement is over" and "ends his movement" is the same thing but what's lacking is the term "phase" to make it crystal clear.

"Evading Monsters
Locations and street areas that contain monster markers
may affect an investigator’s movement. Each time an
investigator attempts to leave a location or street area
occupied by one or more monster markers, the investigator
must either fight or evade each of the monsters there
(see “Evading Monsters” and “Combat” on pages 14 for
more information). Likewise, if an investigator ends his movement in a location or street area occupied by one or
more monsters, he must fight or evade each such monster."

Here if an investigator "ends his movement" (declares it ended?) (and the same phrase required to pick up clues above), then monster engagement begins. But since the investigator "ends his movement) according the first rules quote, s/he can pick up clues.

From the old "Kevin W's answers to rules questions" thread:

"The rules say that an investigator must fight or evade if they end their movement in an area with monsters. They also say an investigator may pick up clues in an area where they end their movement. In an area with both clues and monsters, may they pick up the clues before fighting the monsters, or must they defeat or evade all monsters before getting the clues?

Monsters first, then clues"

I'll spare you my opinion on "ending movement" and "end of movement phase", because it has been discussed enough as of late. We'll need to wait for official word on this.

Morgaln said:

This would be the logical way and apparently the "intention". Thanks. The question seems to be coming up a lot, so it would be nice to have something officially FAQ'd.