Note 7 - Commitments and Contingencies (Details Textual) $ in Millions	Sep. 30, 2022 USD ($)
Number of Operating Leases	1
Lessee, Operating Lease, Term of Contract (Year)	4 years 5 months 1 day
Lessee, Operating Lease, Discount Rate	6.25%
Lessee, Operating Lease, Remaining Lease Term (Year)	2 years 7 months 28 days
ABL Bio [Member]
Milestone Payment Obligation	$ 6

Amount of obligation for milestone payments.

No definition available.

Name:	cmpx_MilestonePaymentObligation
Namespace Prefix:	cmpx_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

The number of operating leases.

+ References

No definition available.

+ Details

Name:	cmpx_NumberOfOperatingLeases
Namespace Prefix:	cmpx_
Data Type:	xbrli:integerItemType
Balance Type:	na
Period Type:	instant

- Definition

Discount rate used by lessee to determine present value of operating lease payments.

+ References

Reference 1: http://www.xbrl.org/2003/role/exampleRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 842
-SubTopic 20
-Section 50
-Paragraph 3
-Subparagraph (c)(3)
-URI https://asc.fasb.org/extlink&oid=128292326&loc=SL77918673-209980

+ Details

Name:	us-gaap_LesseeOperatingLeaseDiscountRate
Namespace Prefix:	us-gaap_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	instant

- Definition

Remaining lease term of operating lease, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 842
-SubTopic 20
-Section 50
-Paragraph 3
-Subparagraph (a)(1)
-URI https://asc.fasb.org/extlink&oid=128292326&loc=SL77918673-209980

+ Details

Name:	us-gaap_LesseeOperatingLeaseRemainingLeaseTerm
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	instant

- Definition

Term of lessee's operating lease, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 842
-SubTopic 20
-Section 50
-Paragraph 3
-Subparagraph (a)(3)
-URI https://asc.fasb.org/extlink&oid=128292326&loc=SL77918673-209980

+ Details

Name:	us-gaap_LesseeOperatingLeaseTermOfContract
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	instant

- Details

Name:	srt_CounterpartyNameAxis=cmpx_ABLBioMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: