No, you cannot start one action, perform a second action, then complete the first action - each action is a discrete portion of time that can't be mixed around like that.
While there is no explicit rule stating that you cannot mix actions, there are specific powers that imply that each action must be performed before starting another action.
A prime example of this is the 13th level Monk Full Discipline power Dance of the Stinging Hornet (Full Discipline powers are - if both techniques are used - effectively the same as any other character using a move action encounter power and a standard action encounter power in the same round - the Monk expends two actions but only has to use one power to gain the benefits on both actions). The movement technique of this power is as follows:
Effect: You make an Athletics check to jump with a +5 power bonus. You are considered to have a running start. You can use the attack technique at any point during this
A number of other Monk powers offer similar functionality by allowing them some type of movement and the use of the associated attack technique at any point during that movement.
It is also worth noting that across the entire range of official 4e books there has never been a power published that states something along the lines of "attacks made during this movement gain x" without also stating that the character gains a specific attack from the power.
These could be considered specific cases of granting attacks during movement, which would imply that it is otherwise not possible.
Some GM's however do allow free actions (or potentially a subset of available free actions) to be taken essentially at the same time as other actions, because oftentimes they have no direct consequence on how a turn plays out (such as yelling something to an ally).